Showing that a space is Banach

Question

I've been given the following problem to solve:

Let $X$ be the space of all functions on $[0,1]$ which vanish at all but a countable number of points and for which \begin{align} \| x \|= \sum_{n=1}^\infty |x(t_n)| < \infty, \end{align} where $t_n$ are the points at which $x$ doesn't vanish.

I'd appreciate a hint on how to proceed to show that the space is complete. That is, that every Cauchy sequence converges to a function in this space.

Answer 1

OK, so the first thing to do is start with an arbitrary Cauchy sequence $x_n \in X$, and construct a limit $x \in X$ from it. You need to think about:

1. How to construct a function $x$ from the sequence $x_n$,
2. How to prove this constructed function $x$ lies in $X$, and
3. How to prove that $x_n \to x$.

For 1, make a guess, using the completeness of $\Bbb{R}$ (i.e. using Cauchy sequences in $\Bbb{R}$), as to what $x$ should do to an arbitrary $t \in [0, 1]$.

For 2, take your construction from part 1, and use the fact that, if you union all the points $t \in [0, 1]$ where $x_n(t) \neq 0$ for some $n$, this is still a countable set.

For 3, well, it's hard to give a hint without spoiling 1. Just try using the definitions.