Let $(u_{n})$ be a convergent sequence converging to $u_{0}$. I want to show that $(|u_{n}|)$ converges to $|u_{0}|$.
Proof. We want to show that given $\epsilon > 0$, $\exists N \in \mathbb{N}$ such that $n \geq N \Rightarrow ||u_{n}| - |u_{0}|| < \epsilon$.
We know that $||u_{n}| - |u_{0}|| < \epsilon \Rightarrow -\epsilon < |u_{n}| - |u_{0}| < \epsilon \Rightarrow |u_{0}| - \epsilon < |u_{n}| < |u_{0}| + \epsilon$. Showing that $|u_{n}| < |u_{0}| + \epsilon$.
My question is, how can we deduce, from the last statement, what $N$ would be? Because the last statement, to me, isn't looking like it converges.
Thanks.