Under choice and dependent choice, a finite subordinate partition of unity for a two-set cover of a compact interval

On $[0,1]$, take the open-in-the-subspace cover $U=(-1,\tfrac34)\cap[0,1]$ and $V=(\tfrac14,2)\cap[0,1]$. Define $\varphi(x)=\max\{0,\min\{1,2-3x\}\},\qquad\psi(x)=1-\varphi(x).$ Then $\varphi$ and $\psi$ are continuous, nonnegative, and sum to one. Their supports are contained respectively in $[0,\tfrac23]\subseteq U$ and $[\tfrac13,1]\subseteq V$, so they are a finite subordinate partition of unity.

The explicit pair is an instance of the existence theorem Under choice and dependent choice, every open cover of a compact Hausdorff space admits a finite subordinate partition of unity, under its stated Choice and Dependent Choice hypotheses.