Definition: Let's call sequence $\{d_i\}_{i=1}^n$ graphic if $d_1,d_2...,d_n$ can be used as a degree of the vertex of a graph.
Now, prove that $d_i$ is graphic iff for every $1\le k\le n$ \[\sum_{i=1}^kd_i\le k(k-1)+\sum_{i=k+1}min(d_i,n)\] and $\sum\limits_{i=1}^nd_i$ is even.
Erdos-Gallai Theorem
One one thing is neutral in the universe, that is $0$.