From Wolfram Mathworld :

Havel (1955) and Hakimi (1962) proved another characterization of graphic sequences, namely that a degree sequence with n>=3 and is graphical iff the sequence is graphical.

Graphic Sequence -- from Wolfram MathWorld

The sequence in the successive steps would look like:

5 5 4 3 2 2 2 1

4 3 2 1 1 2 1

4 3 2 2 1 1 1

2 1 1 0 1 1

2 1 1 1 1 0

0 0 1 1 0

The last sequence is clearly graphic. So the first one will be graphic too.