Prove that an $n$-vertex graph with $5n−10$ edges has a vertex of degree at most $9$

Suppose that $$G$$ is a graph with $$n$$ vertices and $$5n−10$$ edges. Prove that $$G$$ has a vertex of degree at most $$9$$.

I am stuck on how to approach this.

I see that there are two cases:

1. $$n \le 10$$, in this case no vertex can have a degree higher than $$9$$.
2. $$n > 10$$.

I cannot figure out how to prove the problem for the second case.

If it matters, we are working on planar graphs in class right now.

Hint: The sum of the degrees is twice the number of edges, so $$\sum \text{deg}(v) = 2(5n - 10) = 10n - 20$$.