Theorem (Approximation of compact operators) Let $X,Y$ be Banach spaces and $F: X \supset M \to Y$, where $M \ne \emptyset$, be compact. Then for every $n \in \mathbb{N}$ there exists a continuous $F_n: M \to Y$ such that $\max_{v \in M} \| F_n v - F v \| \le \frac{1}{n}$.
Proof. Let $n \in \mathbb{N}$ and $(v_k)_{k = 1}^{m}$ be a finite $\frac{1}{n}$-net for $F(M)$. Then we have $$ \min_{j = 1}^{m} \| F u - v_j \| \le \frac{1}{n} \qquad \forall u \in M. $$ For $u \in M$, $n \in \mathbb{N}$ and $j \in \{1, \ldots, m\}$ let \begin{equation} a_j(u) := \max\left(0, \frac{1}{n} - \| F u - v_j \|\right) \ge 0, \qquad F_n u := \frac{\sum_{j = 1}^{m} a_j(u) v_j}{\sum_{j = 1}^{m} a_j(u)}, \end{equation} which are well-defined and continuous, as is $u \mapsto \| F u - v_j \|$.
For $v \in M$ we have \begin{align} \| F_n v - F v \| & = \left\| \left( \sum_{j = 1}^{m} a_j(v) \right)^{-1} \sum_{j = 1}^{m} a_j(v) v_j - F v \right\| \\ & = \left\| \left( \sum_{j = 1}^{m} a_j(v) \right)^{-1} \sum_{j = 1}^{m} a_j(v) (v_j - F v) \right\| \\ & \le \left( \sum_{j = 1}^{m} a_j(v) \right)^{-1} \left( \sum_{j = 1}^{m} a_j(v) \right) \| v_j - F v \| < \frac{1}{n} \end{align} My question In the last line there is a $v_j$ outside of any summations over $j$, so there must be at least a typo. But I still don't understand, we have that $\min_{j = 1}^{m} \| F v - v_j \| \le \frac{1}{n}$, so how can $\| v_j - F v \|< \frac{1}{n}$ for any $j \in \{1, \ldots, m\}$?
Here's a shorter formulation of the proof, which unfortunately doesn't clear up my misunderstanding.
One variant to repair this and to obtain the strict inequality directly is to pick an index $j^*$ with $\|Fu-v_{j^*}\|=\min_j\|F_u-v_j\|<\frac1n$. Then single this index out in the summation \begin{align} \|Fu-F_nu\|\le... &\le\left(\sum_{j:a_j(u)>0}a_j(u)\right)^{-1}\left(\sum_{j:a_j(u)>0}a_j(u)\|Fu-v_j\|\right)\\ &\le\left(\sum_{j:a_j(u)>0}a_j(u)\right)^{-1}\left(a_{j^*}(u)\|Fu-v_{j^*}\|+\frac1n\sum_{j\ne j^*:a_j(u)>0}a_j(u)\right)\\ &=\frac1n-\left(\sum_{j:a_j(u)>0}a_j(u)\right)^{-1}a_{j^*}(u)\left(\frac1n-\|Fu-v_{j^*}\|\right) <\frac1n \end{align}