Take a normal 6-sided die.
Let event A be "roll any of the numbers 1, 2, 3 or 4". P(A) = 4/6
Let event B be "roll any of the numbers 1 or 2". P(B) = 2/6
P(A) + P(B) = 4/6 + 2/6 = 6/6 = 1
But B is not the complement of A.
The complement of A is the event "roll any of the numbers 5 or 6".
By this example, we've shown that
P(A) + P(B) = 1 does not imply that A and B are complements.
It also seems you have misunderstood the question.
How could it be proven that B isn't the complement of A?
This is not what you need to prove, and you cannot prove it just by knowing the probabilities. What you need to show is that it isn't always the case. You can show this by giving 1 counterexample, as above.