Find a closed walk in a graph that visits every vertex at least once.

Question

Prove that, for all finite connected graph, it is possible to find a closed walk that visits every vertex of the graph at least once.

I have no idea. I've try to prove using minimum spanning tree, but I need a closed walk. Any idea?

Answer 1

Enumerate you vertices $v_1,\dots,v_n$. Because $G$ is connected, you can find a path from $v_i$ to $v_{i+1}$ for every $i=1,\dots,n-1$. So you construct a path starting at $v_1$, visiting $v_2$, $v_3$, etc., and ending at $v_n$. Now you just go back to $v_1$ to close your tourist tour of the graph.