= Urysohn two-thirds approximation
{c}
{title2=$\|h\|_\infty\leq M/3,\quad\|g-h|_C\|_\infty\leq2M/3$}
Let $C$ be closed in a <normal topological space> and $g:C\to[-M,M]$ continuous. Separate its closed level sets $g\leq-M/3$ and $g\geq M/3$ by <Urysohn's lemma> to obtain $h:X\to[-M/3,M/3]$ taking the corresponding endpoint values. Checking the outer and middle ranges gives the displayed bounds. Iteration on residuals gives a uniformly convergent series with total <supremum norm> at most $M$. The <uniform limit theorem> gives the bounded <Tietze extension theorem>.
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