Any bipartite graph has a matching that covers each vertex of maximum degree I need to prove following lemma:

Any bipartite graph has a matching that covers each vertex of maximum
  degree

Any help will be appreciated.

If the maximum degree is $k$, it is possible to keep adding edges and vertices until the graph is $k$-regular. Then by Hall's theorem $G$ will have a perfect matching. If you remove from the matching all the added edges and the isolated vertices you get a matching with the desired property. 
