Let $S=\{a_1,a_2,\dots,a_n\}$ be a set of distinct integers. Let $k$ is the least common multiple of the numbers $\{a_1+1,\dots,a_n+1\}$.
Prove that the sequence that is take all the elements of $S$ $k$ times each is a graphic sequence. Trying to use the Erdos-Gallai theorem but cant prove the second statement with the inequality i try to use induction but i cannot do the second inductive step.
Definition :Graphic sequence if obeys at
You need no induction for this.
By the definition of $k$ you can find $b_i$ such that $b_i(a_i+1)=k$ for all $i$.
Now just observe that the graph $b_1K_{a_1+1}+\ldots+b_nK_{a_n+1}$ has the desired degree sequence (this notation means: $b_1$ copies of $K_{a_1+1}$, $b_2$ copies of $K_{a_2+1}$, etc, all disjoint).