# Linear Isomorphism And Completeness

I am just a beginner. So please be patient with me.

Consider $X=\{\frac 1 n : n \in \mathbb N\} \cup \{0\}$. Prove that the vector space of continuous functions on $X$ is linearly isomorphic to the space of convergent sequences in $\mathbb R$. Using this result, conclude that the set of convergent sequences in the NLS of all bounded real sequences under the sup norm is complete.

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$\bf Hint:$ If $f: X\to\mathbb R$ is continuous then $\lim_{n\to\infty}f(\frac 1 n)=f(0)$ hence $f$ is a convergent sequence. On the other hand, if $(a_n)$ converges to $a$ then the function $f: X\to\mathbb R$ given by $\frac 1 n\to a_n$ and $0\to a$ it is easily seen to be a continuous function.