Integer as Sum of Polygonal Numbers

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then $n$ is:
 * $(1): \quad$ Either triangular or the sum of $2$ or $3$ triangular numbers
 * $(2): \quad$ Either square or the sum of $2$, $3$ or $4$ square numbers
 * $(3): \quad$ Either pentagonal or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers
 * and so on.