Relation between topological closure and algebraic closure

Suppose $\displaystyle ((G , \cdot), \tau_G)$ is a topological group, i.e., a group with a topology and the multiplication and inverse operators are continuous. Let H be a subgroup of G. I can prove that H is a topological group. The problem is with $\displaystyle \overline{H}$.

Denote $\displaystyle \overline{H}$ as the topological closure of H and $\displaystyle \tau_{\ \overline{H}}$ as the subspace topology on $\displaystyle \overline{H}$. Munkres claims in Ch2, exercise 3 that $\displaystyle ( ( \overline{H} , \cdot ) , \tau_{ \ \overline{H}} )$ is a topological space and I'm trying to prove it. If I can prove $\displaystyle \overline{H} $ is a subgroup of G, I can finish the proof (that the multiplication operator and inverse operators in $\displaystyle \overline{H}$ are continuous).

All we have to work with is the structure induced on $\displaystyle \overline{H}$ by the general topological group G. If this were a metric space, we could take convergent sequences and work with that. However, in a general topological space, we might work with topological convergence; if I take neighborhoods of a point of closure, I can get a sequence, but without compactness I don't see how to get a convergent sequence to the point of closure.

Ideas?