This is part of an old analysis prelim I'm reviewing to study for my own prelim.
We define $d_\mathbf{S}$ as: $$ d_\mathbf{S}(x,y) = \sum_{k=1}^\infty \frac1{2^k}\cdot\frac{|x_k-y_k|}{1+|x_k-y_k|} $$
We define $K := \{(x_1,x_2,\cdots)\}\in\mathbf S : |x_i|\leq 1\ \forall i\in\mathbb N$.
We are given that $(\mathbf S,d_\mathbf S)$ is complete. I think K is compact w.r.t. $d_\mathbf S$, but I can't see how to prove it. I can't even settle on what broad approach I think is best. Should I:
- Try to show that every sequence in K has a convergent subsequence with a limit in K,
- Try to show that every open cover of K has a finite subcover,
- Try to show that K is totally bounded?
- Use some other test for compactness?
Any insight would be appreciated.