I hope that this does not disappoint, but I am not giving you more that hints.

In a simple graph of order , the maximum vertex degree is .

Thus any degree sequence if distinct values must be: .

But, if the graph has a vertex of degree zero, we can drop it.

Recall that the sum of the degrees is twice the number of edges.

Try to find a contradiction in all that.

2) Follows trivially from #1.