Fact: There exists a finitely generated non-Hopf group.  (An example is the Baumslag-Solitar group $B(2,3)$, although we cannot prove it yet.)  So, by Lemma 5, there is a finitely generated non-residually finite group.  Thus, free groups are not ERF: if $G$ is a finitely generated non-residually finite group, then Lemma 4 implies that the kernel of a surjection $F_r \rightarrow G$ is not separable in $F_r$.  However, finitely generated subgroups of free groups are separable:

Marshall Hall’s Theorem (1949): $F_r$ is LERF.

This proof is associated with Stallings.

Proof: As usual, let $F_r = \pi_1(X)$ where $X$ is a rose. Let $X' \rightarrow X$ be a covering map with $\pi_1 (X')$ finitely generated.  Let $\Delta \subset X'$ be compact. We need to embed $\Delta$ in an intermediate finite-sheeted covering.

Enlarging $\Delta$ if necessary, we may assume that $\Delta$ is connected and that $\pi_1 (\Delta) = \pi_1(X')$.  Note that we have $\Delta \subset X' \rightarrow X$. By Theorem 5 (see below), the immersion $\Delta \rightarrow X$ extends to a covering $\hat{X} \rightarrow X$. Then $\pi_1 (X') = \pi_1(\Delta) \subset \pi_1 (\hat{X})$. So $X' \rightarrow X$ lifts to a map $X' \rightarrow \hat{X}$.

The main tool in the proof above is this:

Theorem 5: The immersion $\Delta \rightarrow X$ can be completed to a finite-sheeted covering $\hat{X} \rightarrow X$ into which $\Delta$ embeds:

Proof: Color and orient the edges of $X$. Any combinatorial map of graphs $\Delta \rightarrow X$ corresponds uniquely to a coloring and orientation on the edges of $\Delta$.  A combinatorial map is an immersion if and only if at every vertex of $\Delta$, we see each color arriving at most once and leaving at most once. Likewise, it’s a covering map if and only if at each vertex, we see each color arriving exactly once and leaving exactly once.

Let $k$ be the number of vertices of $\Delta$.  For each color $c$, let $k_c$ be the number of edges of $\Delta$ colored $c$. Then there are $k-k_c$ vertices of $\Delta$ missing “arriving” edges colored $c$, and there are $k-k_c$ vertices of $\Delta$ missing “leaving” edges colored $c$.  Choose any bijection between these two sets and use this to glue in $k-k_c$ edges colored $c$. When this is done for all colors, the resulting map $\Delta \subset \hat{X} \rightarrow X$ is clearly a covering.

Note that the proof in fact gives us more.  For instance:

Exercise 6: If $H$ is a finitely generated subgroup of $F_r$, then $H$ is a free factor of a finite-index subgroup of $F_r$.

Exercise 7 (Greenberg’s Theorem): If $H\triangleleft F_r$ and $H$ is finitely generated, then $H$ is of finite index in $F_r$.