Below is the proof:
The only part that I am stuck on is the justification for why $x_n$ is an element of $\ell_\infty$ which is the set of all bounded sequences. I think what the author is trying to show here is that for any coordinate j, there is a k large enough such that $x_j-x^k_j$ is less than one and hence bounded, but how do we know this works for the entire infinite sequence? I thought this can only be done for finite tuples, so just because this is true for the $j$ coordinate, there may be some coordinates that need an even larger $k$ so no one $k$ words for all coordinates.
Best Answer
We start with the inequality $$ |x_j| \leq 1 + \|x^{(k)}\|_\infty. $$ You are right in that this inequality alone does not suffice as $k$ can be arbitrarily large. Our goal then is to show that the right-hand side is bounded by some fixed constant, independent of $k$. The author notes that it is possible to show that the sequence $\{\|x^{(k)}\|_\infty\}$ is Cauchy. Hence, the value of $\|x^{(k)}\|_\infty$ must also be bounded (all Cauchy sequences are bounded), say by some $M$. And so we achieve our goal: $$ |x_j| \leq 1 + \|x^{(k)}\|_\infty \leq 1 + M. $$