Hope this is correct...
First, notice that saying f≤∗g isn't holds means there exists infinite values y0,y1... such that g(yn)<f(yn).
For each n, let x(n) be the smallest x greater than n. the image of n -> x(n) is an infinite set.
we define h like this : h(n)=g(x(n))
We know that there exist f in F such that f(y0)>h(y0); f(y1)>h(y1).... (yi are values in N)
since f(y0)>h(y0) then f(y0)>g(x(y0)) AND we have x(y0)≥y0, plus we know that f is increasing, so f(x(y0))≥f(y0)>g(x(y0)), so we proved that f(x(y0))>g(x(y0)), and we can do the same thing for all x(yn) wich are infinitely many
(sry for mistakes english isn't my native language)