[Munkres, Section41, Ex5] Let $X$ be paracompact. We proved a "shrinking lemma" for arbitrary indexed open coverings of $X$. Here is an "expansion lemma" for arbitrary locally finite indexed families in $X$.
Lemma. Let $\{ B_ \alpha \} _ { \alpha \in J}$ be a locally finite indexed family of subsets of the paracompact Hausdorff space $X$. Then there is a locally finite indexed family $\{ U_ \alpha \} _ { \alpha \in J}$ of open sets in $X$ such that $ B_ \alpha \subset U_ \alpha$ for each $\alpha$.
In the proof of the shrinking lemma(Munkres lemma41.6), the author uses the collection of all open sets such that the closure is contained in the given set. So I used the collection of all open sets containing $B_ \alpha$ for some $ \alpha$. But this seems to not work. How can I prove this?