Convergence of supremum norms I'm studying the difference between pointwise and uniform convergence for sequences of continuous functions.
Is it true that if $f_{n} \to f$ pointwise, then $ \|f_{n}\|_{\infty} \to \|f\|_{\infty}$?
It seems true when they converge uniformly, but I can't find a counterexample if the convergence is only pointwise.

Hint: try with $f_n=$ a function that is zero except in a very narrow and tall spike, narrower and taller with $n$.
