[Math] A Cauchy sequence in a normed vector space which is known to have a convergent subsequence must itself converge.

cauchy-sequencesreal-analysissequences-and-seriesvector-spaces

I am having trouble with the following proof:

Prove that a Cauchy sequence in a normed vector space which is known to have a convergent subsequence must itself converge.

I think that the proof should start something like this:

Let there be a convergent subsequence ($\{p_n\}$) of a cauchy sequence ($\{q_n\}$) in $V$. Suppose that $\{q_n\}$ does not converge in $V$. Then there is no $\varepsilon>0$ such that $||q_n-\vec{L}||<\varepsilon$ for all $n>N$ where $N$ is an integer.

Best Answer

Hint: Let $(x_n)_n$ be a Cauchy sequence and $(x_{n_k})_k$ be a subsequence converging to $x$.

  1. Pick $N$ such that for all $n,m \geq N$, $\Vert x_n - x_m \Vert < \epsilon / 2$.
  2. Pick $K$ such that for all $k \geq K$, $\Vert x_{n_k} - x \Vert < \epsilon / 2$.
  3. Now, let $M = \max\{N,K\}$. For all $n \geq M$, $\Vert x_n - x \Vert \leq \ldots$ (can you finish the rest?)
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