If lim(an + bn) = 0, show that lim(an) = 0 and lim(bn) = 0...why is this true

Let {an} and {bn} be nonnegative sequences. Show that if:

lim [an + bn] = 0, lim (an) = 0 and lim (bn) = 0-------------limit as n goes to infinity for all.

I think I understand, but I am not sure how to formalize it. Since we have nonnegative sequences, I think this implies the limits cannot be negative (and therefore must be greater than or equal to 0), which and the only way to have this true is if the limit of each is zero (so the sum of the limits can be zero)...can someone help me understand this?

Prove this by contradiction.

Suppose that at least one of lim (an) and lim (bn) > 0, possibly infinite (since the sequences are never negative). Then, lim(an + bn) = lim (an) + lim(bn) > 0, which is a contradiction.

I hope this helps!