Let $S_4(n)$ be the number of ways of representing $n$ as a sum of 4 integer squares and suppose I can give you a proof of the result that $\sum_{i=1}^nS_4(i) \sim \frac{\pi^2}{2}n^2$.
Is this result enough to deduce that beyond some $N$, every integer can be represented as the sum of 4 squares (given that we are "weakly" suggesting that $\sum_{i=1}^nS_4(i) >> n$)?