TITLE: Prove that if there is a walk from u to v then there is also a path from u to v.
QUESTION [9 upvotes]: Let G be a graph and let u and v be two of its vertices. Prove that if there is a walk from u to v then there is also a path from u to v.
Using induction on length of a path, how can I solve this?
walk: A walk in a graph G = (V,E) is a sequence of vertices $v_1$, $v_2$, $v_3$ ... $v_k$ s.t. {$v_i$,$v_i$+1} ∈ E for i = 1, ..., k-1
path: walk with no repeated vertices or # of edges

REPLY [2 votes]: Without induction: $(v_1,\ldots,v_k)$ is a walk from $u$ to $v$ if and only if $(w_1,\ldots,w_k)$ is a walk from $v$ to $u$, where $w_i=v_{k+1-i}$ for every $1\leqslant i\leqslant k$.