Stuck on Mathematical Induction Proof

I have the following question: Prove with mathematical induction that $3^n+4^nle 5^n$ for all $nge 2$.
$$text{Assume true: }3^n+4^n le 5^n text{. Prove that $3^{n+1}+4^{n+1} le 5^{n+1}$} \
= 3cdot3^n+4cdot4^n \ =3cdot3^n+4^n(3+1)\
=3cdot3^n+3cdot4^n+4^n \
=3(3^n+4^n)+4^n \
le 3(5^n)+4^n \
(5-2)(5^n)+4^n \
5^{n+1} – 2cdot5^n+4^n\
5^{n+1} – 2cdot(3^n+4^n) +4^n$$
I am stuck on. I see now that I double the $5^n$, and this leads me nowhere. Where can I go from here to solve this? I have tried playing with the algebra, but I am not getting anywhere with this last step.