= Uniform convergence at moving evaluation points
{title2=$f_n(x_n)\to f(x)$}
Suppose <continuous functions> $f_n$ converge by <uniform convergence> to $f$ on a <domain of a function> $X$, and $x_n\to x$ with all these inputs in $X$. The <uniform limit theorem> makes $f$ a <continuous function>, and the <triangle inequality> gives
$$
|f_n(x_n)-f(x)|\leq\|f_n-f\|_\infty+|f(x_n)-f(x)|\longrightarrow0.
$$
The first term handles the changing approximation and the second handles the changing input. For $f_n(x)=x^n$ on $[0,1]$, the inputs $x_n=2^{-1/n}$ instead give $x_n\to1$, $f_n(x_n)=1/2$, and $f(1)=1$. <Pointwise convergence> cannot supply the first uniform bound, and this <pointwise limit> also lacks <continuity> at one.
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