Graph Theory Help on Proofs

Could someone help me improve or make these proofs stronger for the
following two questions. Thanks for the help.

question:
Let G be a graph with average degree d. Then there exists a subgraph of
G with minimum degree at least d/2.

proof:
If the minimum degree of G is at least d/2, one is done. Otherwise,
choosing a vertex v in V(G) with deg(v) < d/2, the induced subgraph G-v
has average degree at least d. Continuing with this procedure if
necessary, will result in the desired subgraph.

question:
Let G be a graph with average degree d. Then there exists a bipartite
subgraph of average degree at least d/2.

proof:
If the minimum of G is at least d/2, then we are done. Otherwise,
choosing a random vertex in G, the bipartite subgraph G-v has average
degree at least d. Continue with process until result is achieved.

Thanks for the help.

.

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