Graph $G$ that does not contain a loop contains a bipartite spanning subgraph $H$, such that for any vertex $v$, there is $d_H(v)\ge\frac{1}{2}d_G(v)$

Question

How do you prove that for any graph $G$ that does not contain a loop* contains a bipartite spanning subgraph $H$, such that for any vertex $v$, there is $d_H(v)\ge\frac{1}{2}d_G(v)$?

^{* A loop is defined as an edge whose both endpoints are the same vertex. Graph $G$ may contain repeated edges.}

Answer 1

Hint: Just take the bipartite subgraph with maximum number of edges.