Proving that every connected graph has a vertex whose removal will not disconnect the graph.

Question

I have not done much proofs before this and need some guidance. I know that for a simple graph such as this :

node - node - node -node

Removing the first and last vertex will not disconnect the graph. So far, I can think of two cases: Where there are leaves, or nodes with degree of 1. Where there are no leaves, which means all nodes have degrees 2 or more.

In the first case, removing the leaf will not disconnect the graph. In the second case, removing any should not disconnect the graph. Problem is, I don't know how to formulate this into a good proof.

Answer 1

Remove a leaf of a spanning tree.

Answer 2

Choose two vertices $u,v$ which maximize the distance $\operatorname d(u,v)$. If $w$ is any other vertex of the graph, the shortest path from $v$ to $w$ does not go though $u$ (else $v$ and $w$ would be farther apart than $v$ and $u$), so deleting $u$ leaves a connected graph.