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Convergence of Cauchy Sequences?
Proof the following theorem:
Theorem: Let a_k be a Cauchy sequence with a_k>0, for all k. Every sequence of this form converges.
2 Answers
- Sean HLv 59 years agoFavorite Answer
This is a strange question because it doesn't seem that the hypothesis a_k>0 is required. Here is a sketch of the proof:
1. Since the sequence is Cauchy it is bounded.
2. Since it is bounded it has convergent subsequence, say a_{k_l} converges to a as l -> infinity.
3. Now use the Cauchy property to prove that a_k -> a as k-> infinity as well.
- ?Lv 44 years ago
To teach this may be a Cauchy sequence, you ought to teach that for ?<0 there is an N such that |X(n) - X(m)| < ? if n and m are > N. Given ? > 0, locate N such that a million/N < ?/2. enable N < n < m Then seem at |X(n) - X(n+a million) + X(n+a million) - X(n+2) + ....- X(m) | <= a finite sum of ameliorations < ?