Uniform convergence preserves boundedness

Uniform convergence preserves boundedness: Let $X$ be a set and let $f_n,f:X\to\mathbb{R}$ (or into a normed space). If $f_n\to f$ uniformly on $X$ and some $f_N$ is bounded on $X$, then $f$ is bounded on $X$ (and hence $f_n$ is bounded for all sufficiently large $n$).

More explicitly, if $\|f_N\|_\infty<\infty$ and $\|f-f_N\|_\infty<\infty$, then $\|f\|_\infty \le \|f_N\|_\infty + \|f-f_N\|_\infty < \infty.$

This lemma is often used to guarantee that uniform limits live in the same “bounded function space” and to justify taking sup norms.

Examples:

• If $f_n(x)=\frac{x}{n}$ on $\mathbb{R}$, then $f_n\to 0$ pointwise but not uniformly on $\mathbb{R}$; indeed boundedness of the limit does not force boundedness of the approximants without uniform control.
• On a bounded domain, uniform convergence plus boundedness of some $f_N$ ensures boundedness of the limit.