Let $M$ be a smooth manifold (without boundary if you want). Let $K$ be a compact subset of $M$. Let $\{(U_\alpha,\phi_\alpha)\}_{\alpha=1}^s$ be a finite number of charts of $M$ such that $\{U_\alpha\}$ covers $K$.
Can I find a smooth partition of unity subordinate to $\{U_\alpha\}$? (i.e. a finite family $\{\rho_\alpha\colon M\to \mathbb{R}\}_{\alpha=1}^s$ of smooth functions such that supp $\rho_\alpha \subseteq U_\alpha$ , im $\rho_\alpha\subseteq[0,1]$, $\{$supp$\rho_\alpha$$\}$ is locally finite and for each point $p\in K$ I have $\sum_\alpha \rho_\alpha(p)=1$ ?
I was thinking of "completing" $\{U_\alpha\}_{\alpha=1}^s$ to an open cover $\{U_\alpha\}_{\alpha\in I}$ of all of $M$ (by adding other charts of my atlas, for example), then taking a smooth partition of unity $\{\rho_\alpha\colon M\to \mathbb{R}\}_{\alpha\in I}$ subordinate to this open cover of $M$, then considering only $\{\rho_\alpha\}_{\alpha=1}^s$. But I'm not shure that then I have $\sum_{\alpha=1}^s \rho_\alpha(p)=1$ for each $p$ in $K$.