Let $M$ be a smooth manifold (at least Hausdorff). Most of the times, as an application of the existence of partitions of unity on $M$, the existence of smooth bump functions is shown (see for example Lee or Warner). It is stated like this:
For any closed subset $A \subseteq M$ and any open subset $U$ containing $A$, there exists a smooth bump function for $A$ supported in $U$, i.e. the bump function is identically equal to $1$ on $A$.
However, it is often usefull to have the following restatement (as far as I can tell from a few proofs):
For any $p \in M$ and any open set $U$ there exists a bump function supported in $U$ and identically equal to $1$ on a neighbourhood of $p$.
So I tried to proof the second statement assuming the first. The proof goes like this: $p$ lies in the domain of a chart, say $(V,\varphi)$. Then $\varphi(U \cap V)$ is open in $\mathbb{R}^n$ and thus contains an open ball $B$. Now we can find a closed ball $K \subseteq B$, centered at $\varphi(p)$, and another open ball $B' \subseteq K \subseteq B$ centered also at $\varphi(p)$. Now we find a smooth bump function supported in $U$ which is identically equal to $1$ in $\varphi^{-1}(K)$, so it is in $\varphi^{-1}(B')$. Is my proof right? Is there any simpler proof?
