Prove that there exists a $k$ such that the sequence obtained by listing each element of $S$ a total $k$ times is a degree sequence of some graph. I'm currently working in the following graph theory excercise:

Let $S = \{2, 6, 7\}$. Prove that there exists a positive integer $k$ such that the sequence obtained by listing each element of $S$ a total $k$ times is a degree sequence of some graph. What is the minimum value of $k$?

I'm basing on the theorem:

But haven't find a way to suit the lemma in my solution, thanks in advance for any hint or help.

Vertex of the degree $7$ has $7$ neighbour, so $k\geq 3$. 
Case $k=3$ is impossible by handshakeing lemma. 
Now try $k=4$:
$$2,2,2,2,6,6,6,6,7,7,7,7$$
Say first vertex with degree 7 is not connected with the 2 degree vertices, then, 
$$2,2,2,2,5,5,5,5,6,6,6$$
and first vertex with degree 6 is not connected with the 2 degree vertices, then, 
$$2,2,2,2,4,4,4,4,5,5$$
and first vertex with degree 5 is not connected with the 2 degree vertices, then, 
$$2,2,2,2,3,3,3,3,4$$
and vertex with degree 4 is not connected with the 2 degree vertices, then, 
$$2,2,2,2,2,2,2,2$$
And with them we make a 8-cycle
