Prove that the function $f(x) = x|\sin(x)|$ gets in the interval$(0,\infty)$ every positive value infinite times.
In other words prove that for every $0<y$ the $f(x) = y$ equation have infinite solutions in the $(0,\infty)$ interval.
I tried to prove the following statement with Intermediate value theorem
and two $x$ sequence that block $y$, $f(a) \le y \le f(b)$ for each value of n and then saying something that for this reason there is a sequence $\{c_n\}$ that make $f(c_n) = y$ for every $n$. How can I prove that?