If a subgroup has finite-intersection with a compact neighbourhood of $1$, then it is discrete? 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.  

Alex's comments, as turned into an answer by the OP, are indeed correct.
Let me just point out that the Hausdorff condition really is the key here.  For instance, let $G$ be any group admitting a nontrivial finite subgroup $H$.  Endow $G$ with the indiscrete topology (in which the only open subsets are $\varnothing$ and $G$): this makes $G$ into a non-Hausdorff topological group.  But the desired conclusion fails: $G$ is a compact neighborhood of the identity which has finite intersection with the subgroup $H$, and $H$ is not discrete (rather it is indiscrete).  
