 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[Not uniformly continuous.] Take $x_n=n$, $y_n=n+1/n$. Then $|x_n-y_n|\to0$, but
$$
|y_n^2-x_n^2|=2+\frac1{n^2}\geq2.
$$
This contradicts the defining implication for <uniform continuity>, for example with $\varepsilon=1$. The <sequential criterion for uniform continuity> makes explicit why arbitrarily close arguments at large distances from the origin matter.

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