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Let $X=\mathrm{Conv}(\mathbf R)$, the collection of all convergent sequences in $\mathbf{R}$. Is the normed space $(X,\|\cdot\|_\infty)$ complete?

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What did you try? – Did Sep 4 '12 at 20:57

HINTS: Suppose that $\langle x_n:n\in\Bbb N\rangle$ is a Cauchy sequence in $X$, where $x_n=\langle x_n(k):k\in\Bbb N\rangle$ for $n\in\Bbb N$. You want to find a sequence $y=\langle y(k):k\in\Bbb N\rangle\in X$ such that $\langle x_n:n\in\Bbb N\rangle\to y$.

  1. Show that for each $k\in\Bbb N$, $\langle x_n(k):n\in\Bbb N\rangle$ is a Cauchy sequence in $\Bbb R$.

  2. Use (1) to get a good candidate for $y$.

  3. Show that $$\lim_{n\to\infty}\|x_n-y\|=0\;.$$

  4. For each $n\in\Bbb N$ let $p_n=\lim\limits_{k\to\infty}x_n(k)$. Show that $\langle p_n:n\in\Bbb N\rangle$ converges to some $p\in\Bbb R$.

  5. Show that $\lim\limits_{k\to\infty}y(k)=p$, and conclude that $y\in X$.

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