combinatorics – handshaking lemma and Erdos-Gallai theorem

The conditions for a sequence to be the degree sequence of a simple graph are given by the Erdos-Gallai theorem in addition to the handshaking lemma. Is there an example of a degree sequence where the handshaking lemma is satisfied, but the Erdos-Gallai theorem is not satisfied and thus, the sequence is not graphic (or vice versa).

Or, does satisfying one of these conditions ensure the other is always satisfied?