Every walk contains a path

Statement

Let $G$ be a graph and let $u, v$ be vertices of $G$. Suppose that there is a walk $W$ from $u$ to $v$. Then there is a path from $u$ to $v$ whose vertices and edges are vertices and edges of $W$.

Remarks

Proof idea: if a walk visits a vertex $x$ twice, delete the closed part between the two visits. Each deletion makes the walk shorter, so the process stops with a path. A consequence: a shortest walk from $u$ to $v$ is a path. Thus the distance can be defined with paths.

Explanations

L1 human

If you can walk from $u$ to $v$, you can also do it without visiting any place twice. Whenever you come back to a place, cut out the loop that you just made.