convergence and boundedness If $f_{n}\to f$ in any norm $\lVert\,\cdot\,\rVert$, then since $\lVert f_{n}\rVert = \lVert f_{n}-f+f \rVert \leqslant \lVert f_n - f\rVert + \lVert f\rVert$, we have that $\lVert f_{n}\rVert$ is uniformly bounded. So one deduces pointwise convergence implies $f_{n}$ is bounded. Yet this is false. Where is the error?
Thanks

The topology of pointwise convergence is - except on finite spaces - not induced by a norm.
However, the argument applies to every continuous seminorm $p$ on a topological vector space and convergent sequence $x_n \to x$, the family $\{ p(x_n) : n\in\mathbb{N}\}$ is bounded.
The topology of pointwise convergence on a space $X$ is induced by the family
$$\bigl\{p_F \colon f \mapsto \sup \{\lvert f(x)\rvert : x \in F\} : F\subset X \text{ finite}\bigr\}.$$
So if $f_n \to f$ pointwise, then $p_F(f_n)$ is bounded for every finite subset $F$ of $X$, and that means the family $\{f_n\}$ is pointwise bounded.
If $X$ is finite, pointwise convergence is the same as uniform convergence, and then a pointwise convergent sequence of functions is uniformly bounded, but if $X$ is infinite, that is generally not the case.
