How to prove that the function $f(t)=1+4t \sin\frac{1}{t}-2\cos\frac{1}{t}$ for $t\ne 0$ takes the value $0$ infinitely often in any neighborhood of $0$?
I was told that this is a corollary of the fact that in any neighborhood of $0$ there are infinitely many segments $I$ such that $t\in I \iff \frac{1}{t}\in[-\frac{\pi}{2}+2\pi k,\frac{\pi}{2}+2\pi k]$. (I'm not sure I'm interpreting it correctly.) I have no clue why this should hold.
Hint: Can you find two sequences $(a_n)$ and $(b_n)$ converging to $0$ and such that
$$0< \dots < a_{n+1} < b_{n+1} < a_n < b_n < \dots < a_2 < b_2 < a_1 < b_1$$
with the property that $f(a_n)>0$ for all $n$ and $f(b_n)<0$ for all $n$?