Let $G=(V,U,E)$ be a bipartite graph, i.e. $V,U$ are sets of vertices s.t. $E\subseteq V\times U$.
It is also given that every vertex (both in $U$ and in $V$) has an even degree.
Prove that there exist some subset $E'\subseteq E\ $ s.t. in the graph $G'=(V,U,E')$ the degree of each vertex is exactly half of its degree in $G$.
Hint: A graph $G$ in which every vertex has even degree can be decomposed into cycles. (That is, $E(G)$ is a disjoint union of cycles.) If $G$ is bipartite, what has to be true about those cycles?