CN-LEAGUE 2008 Problem 2

Let $f(x)$ be a periodic function with periods $T$ and $1$ ( $0<T<1$ ).Prove that: (1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$ ; (2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence { $a_n$ },with $1>a_n>0$ for all positive integer $n$ ,such that all $a_n$ are periods of $f$ .