Prove that $C_0$ (the space of sequences with limit $0$) is complete.
My effort: Let {$x_n$} be sequnce in $C_0$ converging to the limit $0$. As the {$0$} is in $C_0$ hence $C_0$ closed in $C$ which is complete. So $C_0$ so. Is this correct?
Please verify.
To avoid confusing yourself with sequences of sequences, I recommend thinking of the elements of $C_0$ as functions $f:\mathbb{N} \to\mathbb{R}$ such that $\lim_{n\to\infty}f(n)=0$. Then the claim becomes: if $(f_k)_{k=1}^\infty$ is a Cauchy sequence of functions with respect to the uniform norm, there is $f\in C_0$ such that $f_k\to f$ uniformly.
The proof consists of two steps:
For each $n\in\mathbb{N}$ the sequence $(f_k(n))_{k=1}^\infty$ is a Cauchy sequence of numbers, hence has a limit. Denote this limit by $f(n)$, and you have a function $f:\mathbb{N}\to \mathbb{R}$.
To show that $f\in C_0$: given $\epsilon>0$ pick $n$ such that $\|f-f_n\|<\epsilon/2$, and then pick $K$ such that $|f_n(k)|<\epsilon/2$ whenever $k\ge K$. Conclude that $|f(k)|<\epsilon$ whenever $k\ge K$.