Let $F$ be a free group, $H \leq F$ and $[F:H]=n \in \mathbb{N}$. Show that whenever $K \leq F$ then we have $K \cap H \neq 1$

Question

Let $F$ be a free group, $H \leq F$ and $[F:H]=n \in \mathbb{N}$. Show that whenever $K \leq F$ then we have $K \cap H \neq 1$

I'm trying to figure out why this is, I'm having trouble, if somebody could walk me through a proof that would be great. Also, what are the different ways of understanding that every nontrivial element of a free group has infinite order? Thanks in advance!!

Answer 1

Let $x\neq 1\in K$.

Then $x^m\in H$ for some $m\le n $ (because there are only $n$ $H$-cosets).

So $x^m\in H\cap K$.

So $x^m=1$.

So $x$ has finite order in $F$.