double negatives are common in colloquial English.

many Americans say: I don't know nothing.

I am not unconvinced by his argument.

The use of not together with unconvinced suggests that the speaker has a few mental reservations about the argument. The double negative creates a nuance of meaning that would not be present had the speaker just said:
I am convinced by his argument. oxforddictionaries.com

granted, it sounds better when a native speaker use double negatives.