University:
Massachusetts Institute of Technology
Course:
18.100C | Real Analysis
Academic year:

2021
Views:

132

Pages:

1
Author:

Carley Peters

18.100C Lecture 8 Summary Convergent sequences in metric spaces. Examples. Theorem 8.1. Let (xn ) be a convergent sequence, where all the xn lie in a ¯ subset E ⊂ X. Then the limit x lies in E. Theorem 8.2. If x ∈ Ē, there is a sequence (xn ), xn ∈ E, which converges to x. Subsequence of a convergent sequence is convergent (same limit). Theorem 8.3. Let (X, d) be a compact metric space. Then every sequence (xn ) in X has a convergent subsequence. Corollary 8.4. Every bounded sequence in Rd has a convergent subsequence. Deﬁnition of Cauchy sequence. Every convergent sequence is a Cauchy se quence. Lemma 8.5. Let (xn ) be a Cauchy sequence. If it has a convergent subsequence, then (xn ) itself converges (to the same point). Theorem 8.6. Let (X, d) be a compact metric space. Then every Cauchy se quence converges. Corollary 8.7. Every Cauchy sequence in Rn converges. A metric space where this happens (every Cauchy sequence converges) is called complete. So, we just showed that compact metric spaces as well as Rn are complete. 1

18.100C Real Analysis: Lecture 8 Summary

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