The problem I'm trying is to prove whether or not the metric space of real-valued sequences $(x_n)$ such that $x_n=0$ for all but finitely many values of $n$, with the sup metric: $d((x_n),(y_n)) = max |x_n-y_n|$ is complete.
Now, I tried to find counterexamples but did not seem to find any...
Hence I'm trying to prove that if a sequence $((x_n)^{(k)}) : (x_n)^{(1)}, (x_n)^{(2)}, ... $ is Cauchy, then it converges to some sequence.
We can easily show how the sequence of the $k$th entries: $(x_k^{(n)})$ for a fixed $k$ is Cauchy in $\mathbb{R}$ and hence convergent. Moreover for all but finitely many values of $k$ we can find a subsequence of this series which is constantly zero. So for all but finitely many values of $k$, $(x_k^{(n)})\rightarrow 0$ by properties of real Cauchy sequences.
For finitely many values $k_1,k_2,...$, $(x_{k_{i}}^{(n)})\rightarrow \alpha_i$ where the limit could be nonzero.
This shows that $((x_n)^{(k)})\rightarrow (\alpha_n)$ where this limit sequence is still an element of the space we are considering, hence this space is complete.
This is my proof, but I feel the part in which I claim the existence of a constantly zero subsequence needs a little bit more care...
Any help or comment would be extremely helpful!!