How to prove the relation between dimensions of linear subspaces and their vector space?

Question

If $H_1, ... , H_k$ are linear subspaces of $V$ with $k≤n$, so that each $H_i$ has the dimension $n-1$.

How can I prove that $dim(H_1 \cap ... \cap H_k) ≥ n-k$?

I have a feeling this might be doable with the help of induction, but cannot really formulate how to set up the problem.