At page 79 of Basic Number Theory by André Weil, there is an argument showing that a subgroup of a topological group is discrete,
because there is a compact neighbourhood of $1$ with finite intersection with that subgroup $\Gamma$.
So I am wondering how one proves the following:
If a sub-group of a topological Hausdorff group has finite intersection with a compact neighbourhood of $1$, then it is discrete.
Since one has earlier spotted the statement that a discrete subset of a compact set is finite, I think, in this book, by discrete one understands closed discrete.
Now, we know that this subset $\Gamma$ has a discrete intersection with a compact neighbourhood. But how could this fact help us showing the discreteness of the whole subset $\Gamma$? This is where I am stuck.
Any hint is well-appreciated.