I'm trying to prove this theorem: if $\lim _{x\to \infty }\left(f\left(x\right)\:\cdot \:f\left(\frac{13}{5}x\right)\right)\:=\:\infty$ then $\lim _{x\to \infty }f\left(x\right)\:=\:\infty$.
I acctualy tried to disprove it by a counterexample when f(x) is a split function that gives the value 1 for integers and the value of $X^2$ when x is a fraction, but when x goes to infinity it's the same...
Later I thought that it must be true because if the multiplication of a function by itself has limit in infinity infinity, then the function itself should also have limit at infinity, but I don't know how to prove it.
Appreciate the help!