Exercise 3.20

Question

Exercise 20:
    Suppose $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subsequence $\{p_{n_i}\}$ converges to a point $p\in X$. Prove that the full sequence $\{p_n\}$ 
    converges to $p$.

ghostofgarborg · Accepted Answer

\def\eps{\varepsilon} Choose $\epsilon > 0$. Since $ p_n $ is Cauchy, we can choose an $N$ such that $d(p_m, p_n) < \frac \epsilon 2$ whenever $m, n > N$. Assume $p_{n_i} \to p$. Then there is an $M$ such that whenever $i > M$, $d(p_{n_i}, p) < \frac \epsilon 2$. For $n > N$, there is an $i > M$ such that $n_i > N$. Then $$ d(p_n, p) \leq d(p_n, p_{n_i}) + d(p_{n_i}, p) < \frac \epsilon 2 + \frac \epsilon 2 = \epsilon $$ This implies that the sequence converges to $p$.

Exercise 3.20

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Exercise 3.20

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