Lagrange's Four Square Theorem

Theorem
Every positive integer can be expressed as a sum of four squares.

Proof
From Product of Sums of Four Squares it is sufficient to show that each prime can be expressed as a sum of four squares.

The prime number $2$ certainly can: $2 = 1^2 + 1^2 + 0^2 + 0^2$.

Now consider the odd primes.

Suppose that some multiple $m p$ of the odd prime $p$ can be expressed as:
 * $m p = a^2 + b^2 + c^2 + d^2, 1 \le m < p$.

If $m = 1$, we have the required expression.

If not, then after some algebra we can descend to a smaller multiple of $p$ which is also the sum of four squares:
 * $m_1 p = a_1^2 + b_1^2 + c_1^2 + d_1^2, 1 \le m_1 < m$.

Next we need to show that there really is a multiple of $p$ which is a sum of four squares.

From this multiple we can descend in a finite number of steps to $p$ being a sum of four squares.

Also see
Waring's Problem