Understanding Real Analysis: Cauchy Sequences and Convergence
Dive into the world of Real Analysis with a focus on Cauchy sequences and convergence. Learn the definitions, properties, and examples to grasp the fundamental concepts in mathematics that form the basis of advanced analysis.
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DEPARTMENT OF MATHEMATICS SGGSJ GOVERNMENT COLLEGE PAONTA SAHIB
COURSE NAME : REAL ANALYSIS COURSE CODE : 201TH
CONTENTS Cauchy Sequence A Cauchy sequence is bounded Convergence of a Cauchy Sequence A convergent sequence is always a Cauchy sequence Example based on Cauchy sequence
Cauchy Sequence A sequence {xn } is said to be a Cauchy sequence if given >0, however small ,there exist a positive integer k (depending upon ) such that |xn xm| < n,m >k Another Def. A sequence {xn } is said to be a Cauchy sequence if given >0,however small, a positive integer m (depending upon ) such that , |xn+p xn| < n m and p N.
A CAUCHY SEQUENCE IS BOUNDED Let {xn} be Cauchy sequence. given >0, there exist a positive integer p such that |xn-xm|< n,m p (1) In particular, |xn-xm|< n p ...(2) Now, |xn| = |(xn-xp)+ xp| |xn-xm| + |xp| < + |xp| n p [ of (2) ] |xp| < + |xp| n p Let M = max. { |x1|,|x2|, .|xp-1|, +|xp| } {xn} is bounded.
Convergence of Cauchy sequence Let {xn} be a Cauchy sequence . given >0, there exist a positive integer p such that |xn-xm|< n,m p (1) In particular, |xn-xm|< n p ...(2) Now, |xn| = |(xn-xp)+ xp| |xn-xm| + |xp| < + |xp| n p [ of (2) ] |xp| < + |xp| n p Let M = max. { |x1|,|x2|, .|xp-1|, +|xp| } {xn} is bounded.
by Bolzano-Weistrass Theorem, {xn} has a convergent subsequence {xnk} . Let {xnk} be convergent to l. We shall prove tha {xnk } also converges to l. Since xnk l given >0, a positive integer p ,s.t |xnk- l | < k p (3) for n p, nk np p , from (1), we have , |xn- xnk| < /2 | xn-l| (xn- xnk) + ( xnk - l < /2 + /2 n p [ of (3),(4) ] (4)
= np | xn l | < n p { xn} is convergent .
A Convergent sequence is always a Cauchy sequence Let the sequence {xn} converges to l given >0,however small, k N, s.t. | xn- l | < n p (1) Let m k be a natural number . |xm-l| < /2 Now |xn-xm| = | (xn-l)+ (l- xm) | |(xn-l)|+ |(l- xm)|
= |xn - l| + |xm - l| < + |xn - xm| < n,m k {xn} is a Cauchy sequence.
EXAMPLE : Prove that the following sequences are not Caunchy sequences : (i) {(-1)n } (ii) {(-1)n n} (iii) {n2} SOLUTION: (i) Here xn= (-1)n x2n=(-1)2n = 1 , x2n+1=(-1)2n+1 = -1 Let = 1 Now |x2n+1 x2n|= |-1-1|= |-2|= 2>1 = n {xn} is not a Cauchy sequence.
(ii) Here xn=(-1)n n x2n= (-1)2n(2n) = 2n , x2n+1= (-1)2n+1(2n+1) = -(2n+1) Let =1 Now |x2n+1- x2n|= |-(4n+1) |= 4n+1 >1 = n {xn} is not a convergent sequence. (iii) Here xn = n2 , xn+1= (n+1)2 = n2+2n+1 Let =1 |x x | = |n2+2n+1- n2| = |2n+1|=2n+1 > 1 = n {xn} is not a Cauchy sequence.
