Proving periodicity of an arbitrary function

Question

I could not think of a proof to probably trivial question: Let $y=f(x)$ be a periodic function on any closed interval $a\le{x}\le{a+T}$, if T is a period of the function $f(x)$, then the numbers $2T, 3T, 4T,...$ are also periods. Prove validity of $f(x)=f(x+T)=f(x+2T)=f(x+3T)=\cdots$?

Answer 1

Hint: Prove by induction that $(\forall n\in\mathbb N):f(x+nT)=f(x)$.