I was wondering if my proof of the uniform continuity of $f(x) = \inf\{| x - a | \colon a \in S\}, S \subset \mathbb{R}$ was correct.
Attempt:
$ | f(x) - f(y) | = | \inf\{|(x-a)|\} - \inf\{|(y-a)|\} | \leq | \inf(| x - a | - | y - a|) | \leq | \inf(| x - a - y + a|) | \leq | \inf\{| x - y| \} | \leq | x - y | < \delta$
Therefore, by choosing $\delta = \epsilon$, we get:
$| f(x) - f(y) | < \epsilon$, proving the function converges uniformly.