Let a tree with a vertex of degree 4 .prove that there are at least four leaves in the tree

Question

I wrote some proof to the claim above, and don't know if it is correct:

The claim: "Let a tree with a vertex of degree 4 .prove that there are at least 4 leaves in the tree".

My proof: by contradiction, suppose there is a tree T which has vertex with degree 4,

and less than 4 leafs.

Let v be the vertex with 4 degree. we travel from him along the tree, and because there is 3 leafs at most, we arrive to a vertex we alreay visited. If we arrive to such vertex, suppose to be u, it means there are 2 paths for us arrive to u from v. that means, there is a cycle in the tree, which contradicts the claim T is a tree.