Let $(x_n)$ be a sequence of orthogonal vectors in a Hilbert space $(V, \langle,\rangle)$. For $n = 1, 2, 3, ... $ put, $$s_n = \sum_{j=1}^{n} x_j.$$
(a) Calculate $\|s_n\|$ in terms of $\|x_1\|,\|x_2\|,...,\|x_n\|$
(b) Let $n > m$. Calculate $\|s_n - s_m\|$.
(c) Assume that there is $K > 0$ such that $\sum_{j=1}^{n} \| x_j \|^2 \leq K$ for all $n\in\mathbb{N}$. Prove that $(s_n)$ is a Cauchy sequence in V and give a reason why $\sum_{j=1}^{\infty} x_j$ is a convergent series in V.
I can do (a) and (b) without any problems, but I can't seem to figure out c. I have tried going through the definition of a Cauchy sequence, fixing $\epsilon > 0$ and trying to choose an $N$. I have tried splitting it into cases where $\epsilon > K$ and $K > \epsilon$. I feel like I am missing something quite simple, can somebody help me?
Assume such a $K$ exists. Let $\varepsilon>0$, we aim to find $N$ such that $\lVert s_m-s_n\rVert<\varepsilon$ whenever $m,n\geq N$.
We have $\sum_{j=1}^n\lVert x_j\rVert^2\leq K$ for each $n$, so the series $S:=\sum_{j=1}^\infty \lVert x_j\rVert^2$ converges and has $S\leq K$ (as it is a bounded, increasing sequence of positive real numbers). Now there must exist a natural number $N$ for which $S-\sum_{j=1}^{n}\lVert x_j\rVert^2<\varepsilon^2$ for each $n\geq N$. Take $m,n\geq N$ with $n\geq m$. Then, by Pythagoras (for Hilbert spaces), we have $$ \lVert s_n-s_m\rVert^2=\sum_{j=m}^n\lVert x_j\rVert^2\leq \sum_{j=m}^\infty\lVert x_j\rVert^2<\varepsilon^2. $$ It follows that $\lVert s_n-s_m\rVert<\varepsilon$. Thus $(s_n)$ is a Cauchy sequence. I leave you to speculate as to why $\sum_{j=1}^\infty x_j$ converges.