Let $H$ and $K$ be finite index subgroups of a group $G$ with index $h$ and $k$, respectively.
I know that $H\cap K$ is of finite index in $H$ and $K$.
Is the index of $H\cap K$ in $H$ bounded by $k$?
By symmetry the index of $H\cap K$ in $K$ would be bounded by $h$.
In standard notation: does the inequality
$$[H:H\cap K] \leq [G:H]$$ hold? Edit: I meant to write $[H:H\cap K] \leq [G:K]$.
There are many related MS questions, but I don't see how they answer this question.