In my real analysis class, the following lemma was presented to us:
Lemma (Lindelöf): Let $A\subseteq \mathbb{R}^d$. Assume that for each $i\in I$, we have that $O_i\subseteq\mathbb{R}^d$ is open. If $A\subseteq \bigcup_{i\in I}O_i$, then there exists a countable $J\subseteq I$ such that $A\subseteq \bigcup_{i\in J} O_i$.
Question: Why is this lemma useful? The case where $J = I$ seems trivial to me and it's not talking about proper subsets $J\subsetneq I$. If you want to apply this lemma there's always the possibility that the lemma just holds for the trivial case, and you don't know anything new (I'm obviously mistaken).
It is very useful for constructing an approximation of unity fo $A$, i.e. a fonction f such that $f|_A\equiv 1$ and $f(x)=0$ if $d(x,A)>\epsilon$. These kind of constructions are fundamental in differential geometry.