I'm trying to get a grasp on point-wise convergence and am hoping to prove something to give a concrete example of why it's weak. The lemma goes as follows .
Suppose $f _ { n } : [ a , b ] \rightarrow \mathbb { R }$ is a sequence of continuous functions that converge point wise , but NOT UNIFORMLY to a continuous function $ f : [ a , b ] \rightarrow \mathbb { R } $ .
Then there exists a convergence sequence $ x _ { n } \rightarrow x \operatorname { in } [ a , b ] $ such that $f _ { n } \left( x _ { n } \right)$ does not converge to $ f(x) $ .
Since the convergence is not uniform there exists $\epsilon>0$ and a subsequence $f_{n_i}(x_{n_i})$ (where we can assume the $x_{n_i}$ are convergent to some $x$ because $[a,b]$ is compact) such that $|f_{n_i}(x_{n_i})-f(x_{n_i})|\ge \epsilon$. For $i$ large enough, $|f(x)-f(x_{n_i})|<\epsilon/2$. By the Reverse Triangle Inequality: $$ |f_{n_i}(x_{n_i})-f(x)|\ge||f_{n_i}(x_{n_i})-f(x_{n_i})| -|f(x_{n_i})-f(x)||\ge \epsilon/2$$ So $f_{n_i}(x_{n_i})$ cannot converge to $f(x)$.