Formalizing converging norm Let $x \in \ell^2$. Define $(u_N)$ as the sequence of vectors $u_N \in U$ such that $u_n = x_n$ if $n \le N$, $0$ otherwise. 

Then $\lVert u_N \rVert$ is an increasing function which converges to finite $\lVert x \rVert$. 

The statement seems obvious, but how can I make this claim more rigorous? Saying the norm of $u_N$ increases doesn't seem convincing.

I suppose it depends on how norm is defined. If defined as an infinite sum which is defined as the limit of a finite summation, then the result follows by definition ...
$\sqrt {\sum\limits_{k=1}^{n} |x_k|^{2}}$ increases to $\sqrt {\sum\limits_{k=1}^{\infty} |x_k|^{2}}$. This is just deinition of the infinite sum $\sum\limits_{k=1}^{\infty} |x_k|^{2}$.
$$\|x\|^2\geq\sum_{n=1}^{N+1}x_n^2=\|u_{N+1}\|^2=x_1^2+\cdots + x_{N+1}^2\geq x_1^2+\cdots + x_{N}^2=\|u_N\|^2$$