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Type | Label | Description |
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Statement | ||
Theorem | climshft2 13701* | A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.) |
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Theorem | climrecl 13702* | The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.) |
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Theorem | climge0 13703* | A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.) |
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Theorem | climabs0 13704* | Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.) |
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Theorem | o1co 13705* | Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.) |
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Theorem | o1compt 13706* | Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.) |
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Theorem | rlimcn1 13707* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.) |
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Theorem | rlimcn1b 13708* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimcn2 13709* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.) |
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Theorem | climcn1 13710* | Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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Theorem | climcn2 13711* | Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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Theorem | addcn2 13712* | Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 20298 and df-cncf 21965 are not yet available to us. See addcn 21952 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.) |
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Theorem | subcn2 13713* | Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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Theorem | mulcn2 13714* | Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.) |
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Theorem | reccn2 13715* | The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.) |
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Theorem | cn1lem 13716* | A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | abscn2 13717* | The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | cjcn2 13718* | The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | recn2 13719* | The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | imcn2 13720* | The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | climcn1lem 13721* | The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.) |
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Theorem | climabs 13722* | Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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Theorem | climcj 13723* | Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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Theorem | climre 13724* | Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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Theorem | climim 13725* | Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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Theorem | rlimmptrcl 13726* | Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimabs 13727* | Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimcj 13728* | Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimre 13729* | Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimim 13730* | Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | o1of2 13731* | Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | o1add 13732 | The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
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Theorem | o1mul 13733 | The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
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Theorem | o1sub 13734 | The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Fan Zheng, 14-Jul-2016.) |
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Theorem | rlimo1 13735 | Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.) |
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Theorem | rlimdmo1 13736 | A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.) |
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Theorem | o1rlimmul 13737 | The product of an eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.) |
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Theorem | o1const 13738* | A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
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Theorem | lo1const 13739* | A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | lo1mptrcl 13740* | Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1mptrcl 13741* | Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1add2 13742* | The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1mul2 13743* | The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1sub2 13744* | The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) |
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Theorem | lo1add 13745* | The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | lo1mul 13746* | The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | lo1mul2 13747* | The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1dif 13748* | If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | lo1sub 13749* |
The difference of an eventually upper bounded function and an eventually
bounded function is eventually upper bounded. The "correct"
sharp
result here takes the second function to be eventually lower bounded
instead of just bounded, but our notation for this is simply
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | climadd 13750* | Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.) |
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Theorem | climmul 13751* | Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.) |
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Theorem | climsub 13752* | Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | climaddc1 13753* |
Limit of a constant ![]() |
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Theorem | climaddc2 13754* |
Limit of a constant ![]() |
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Theorem | climmulc2 13755* |
Limit of a sequence multiplied by a constant ![]() |
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Theorem | climsubc1 13756* |
Limit of a constant ![]() |
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Theorem | climsubc2 13757* |
Limit of a constant ![]() |
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Theorem | climle 13758* | Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
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Theorem | climsqz 13759* | Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.) |
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Theorem | climsqz2 13760* | Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 14-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.) |
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Theorem | rlimadd 13761* | Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | rlimsub 13762* | Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | rlimmul 13763* | Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | rlimdiv 13764* | Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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Theorem | rlimneg 13765* | Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.) |
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Theorem | rlimle 13766* | Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.) |
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Theorem | rlimsqzlem 13767* | Lemma for rlimsqz 13768 and rlimsqz2 13769. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.) |
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Theorem | rlimsqz 13768* | Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.) |
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Theorem | rlimsqz2 13769* | Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by Mario Carneiro, 3-Feb-2014.) (Revised by Mario Carneiro, 20-May-2016.) |
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Theorem | lo1le 13770* | Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.) |
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Theorem | o1le 13771* | Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
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Theorem | rlimno1 13772* | A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | clim2ser 13773* | The limit of an infinite series with an initial segment removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
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Theorem | clim2ser2 13774* | The limit of an infinite series with an initial segment added. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
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Theorem | iserex 13775* | An infinite series converges, if and only if the series does with initial terms removed. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 27-Apr-2014.) |
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Theorem | isermulc2 13776* | Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) |
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Theorem | climlec2 13777* | Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.) |
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Theorem | iserle 13778* | Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.) |
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Theorem | iserge0 13779* | The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.) |
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Theorem | climub 13780* | The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) |
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Theorem | climserle 13781* | The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
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Theorem | isershft 13782 | Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.) |
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Theorem | isercolllem1 13783* | Lemma for isercoll 13786. (Contributed by Mario Carneiro, 6-Apr-2015.) |
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Theorem | isercolllem2 13784* | Lemma for isercoll 13786. (Contributed by Mario Carneiro, 6-Apr-2015.) |
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Theorem | isercolllem3 13785* | Lemma for isercoll 13786. (Contributed by Mario Carneiro, 6-Apr-2015.) |
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Theorem | isercoll 13786* | Rearrange an infinite series by spacing out the terms using an order isomorphism. (Contributed by Mario Carneiro, 6-Apr-2015.) |
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Theorem | isercoll2 13787* | Generalize isercoll 13786 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.) |
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Theorem | climsup 13788* | A bounded monotonic sequence converges to the supremum of its range. Theorem 12-5.1 of [Gleason] p. 180. (Contributed by NM, 13-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.) |
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Theorem | climcau 13789* | A converging sequence of complex numbers is a Cauchy sequence. Theorem 12-5.3 of [Gleason] p. 180 (necessity part). (Contributed by NM, 16-Apr-2005.) (Revised by Mario Carneiro, 26-Apr-2014.) |
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Theorem | climbdd 13790* | A converging sequence of complex numbers is bounded. (Contributed by Mario Carneiro, 9-Jul-2017.) |
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Theorem | caucvgrlem 13791* | Lemma for caurcvgr 13793. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by AV, 12-Sep-2020.) |
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Theorem | caucvgrlemOLD 13792* | Lemma for caurcvgr 13793. (Contributed by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.) Obsolete version of caucvgrlem 13791 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | caurcvgr 13793* |
A Cauchy sequence of real numbers converges to its limit supremum. The
third hypothesis specifies that ![]() |
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Theorem | caurcvgrOLD 13794* |
A Cauchy sequence of real numbers converges to its limit supremum. The
third hypothesis specifies that ![]() |
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Theorem | caucvgrlem2 13795* | Lemma for caucvgr 13796. (Contributed by NM, 4-Apr-2005.) (Proof shortened by Mario Carneiro, 8-May-2016.) |
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Theorem | caucvgr 13796* | A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Revised by Mario Carneiro, 8-May-2016.) |
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Theorem | caurcvg 13797* |
A Cauchy sequence of real numbers converges to its limit supremum. The
fourth hypothesis specifies that ![]() |
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Theorem | caurcvgOLD 13798* |
A Cauchy sequence of real numbers converges to its limit supremum. The
fourth hypothesis specifies that ![]() |
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Theorem | caurcvg2 13799* | A Cauchy sequence of real numbers converges, existence version. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 7-Sep-2014.) |
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Theorem | caucvg 13800* | A Cauchy sequence of complex numbers converges to a complex number. Theorem 12-5.3 of [Gleason] p. 180 (sufficiency part). (Contributed by NM, 20-Dec-2006.) (Proof shortened by Mario Carneiro, 15-Feb-2014.) (Revised by Mario Carneiro, 8-May-2016.) |
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