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Theorem List for Metamath Proof Explorer - 12301-12400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremo1resb 12301 The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremclimeq 12302* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremlo1eq 12303* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremrlimeq 12304* Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)

Theoremo1eq 12305* Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremclimmpt 12306* Exhibit a function with the same convergence properties as the not-quite-function . (Contributed by Mario Carneiro, 31-Jan-2014.)

Theorem2clim 12307* If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremclimmpt2 12308* Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremclimshftlem 12309 A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)

Theoremclimres 12310 A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremclimshft 12311 A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Theoremserclim0 12312 The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)

Theoremrlimcld2 12313* If is a closed set in the topology of the complexes (stated here in basic form), and all the elements of the sequence lie in , then the limit of the sequence also lies in . (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimrege0 12314* The limit of a sequence of complexes with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimrecl 12315* The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)

Theoremrlimge0 12316* The limit of a sequence of nonnegative reals is nonnegative. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremclimshft2 12317* A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)

Theoremclimrecl 12318* 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.)

Theoremclimge0 12319* A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)

Theoremclimabs0 12320* 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.)

Theoremo1co 12321* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremo1compt 12322* Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremrlimcn1 12323* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremrlimcn1b 12324* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimcn2 12325* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)

Theoremclimcn1 12326* Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremclimcn2 12327* Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremaddcn2 12328* 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 17231 and df-cncf 18847 are not yet available to us. See addcn 18834 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremsubcn2 12329* Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremmulcn2 12330* Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)

Theoremreccn2 12331* The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)

Theoremcn1lem 12332* A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremabscn2 12333* The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremcjcn2 12334* The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremrecn2 12335* The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremimcn2 12336* The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremclimcn1lem 12337* The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremclimabs 12338* 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.)

Theoremclimcj 12339* 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.)

Theoremclimre 12340* 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.)

Theoremclimim 12341* 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.)

Theoremrlimmptrcl 12342* Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimabs 12343* Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimcj 12344* Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimre 12345* Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimim 12346* Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremo1of2 12347* Show that a binary operation preserves eventual boundedness. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremo1add 12348 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.)

Theoremo1mul 12349 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.)

Theoremo1sub 12350 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.)

Theoremrlimo1 12351 Any function with a finite limit is eventually bounded. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremrlimdmo1 12352 A convergent function is eventually bounded. (Contributed by Mario Carneiro, 12-May-2016.)

Theoremo1rlimmul 12353 The product of an eventually bounded function and a function of limit zero has limit zero. (Contributed by Mario Carneiro, 18-Sep-2014.)

Theoremo1const 12354* A constant function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)

Theoremlo1const 12355* A constant function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1mptrcl 12356* Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1mptrcl 12357* Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1add2 12358* The sum of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1mul2 12359* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1sub2 12360* The product of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremlo1add 12361* The sum of two eventually upper bounded functions is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1mul 12362* 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.)

Theoremlo1mul2 12363* 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.)

Theoremo1dif 12364* If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremlo1sub 12365* 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 , so it is just a special case of lo1add 12361. (Contributed by Mario Carneiro, 31-May-2016.)

Theoremclimadd 12366* 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.)

Theoremclimmul 12367* 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.)

Theoremclimsub 12368* 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.)

Theoremclimaddc1 12369* Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimaddc2 12370* Limit of a constant added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimmulc2 12371* Limit of a sequence multiplied by a constant . Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremclimsubc1 12372* Limit of a constant subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)

Theoremclimsubc2 12373* Limit of a constant minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)

Theoremclimle 12374* Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremclimsqz 12375* 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.)

Theoremclimsqz2 12376* 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.)

Theoremrlimadd 12377* Limit of the sum of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimsub 12378* Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimmul 12379* Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimdiv 12380* Limit of the quotient of two converging functions. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremrlimneg 12381* Limit of the negative of a sequence. (Contributed by Mario Carneiro, 18-May-2016.)

Theoremrlimle 12382* Comparison of the limits of two sequences. (Contributed by Mario Carneiro, 10-May-2016.)

Theoremrlimsqzlem 12383* Lemma for rlimsqz 12384 and rlimsqz2 12385. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 20-May-2016.)

Theoremrlimsqz 12384* 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.)

Theoremrlimsqz2 12385* 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.)

Theoremlo1le 12386* Transfer eventual upper boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 26-May-2016.)

Theoremo1le 12387* 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.)

Theoremrlimno1 12388* A function whose inverse converges to zero is unbounded. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremclim2ser 12389* 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.)

Theoremclim2ser2 12390* 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.)

Theoremiserex 12391* 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.)

Theoremisermulc2 12392* Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremclimlec2 12393* Comparison of a constant to the limit of a sequence. (Contributed by NM, 28-Feb-2008.) (Revised by Mario Carneiro, 1-Feb-2014.)

Theoremiserle 12394* Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.)

Theoremiserge0 12395* 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.)

Theoremclimub 12396* The limit of a monotonic sequence is an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 10-Feb-2014.)

Theoremclimserle 12397* 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.)

Theoremisershft 12398 Index shift of the limit of an infinite series. (Contributed by Mario Carneiro, 6-Sep-2013.) (Revised by Mario Carneiro, 5-Nov-2013.)

Theoremisercolllem1 12399* Lemma for isercoll 12402. (Contributed by Mario Carneiro, 6-Apr-2015.)

Theoremisercolllem2 12400* Lemma for isercoll 12402. (Contributed by Mario Carneiro, 6-Apr-2015.)

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