Stuck on Mathematical Induction Proof I have the following question: Prove with mathematical induction that $3^n+4^n\le 5^n$ for all $n\ge 2$.
$$\text{Assume true: }3^n+4^n \le 5^n \text{. Prove that $3^{n+1}+4^{n+1} \le 5^{n+1}$} \\
= 3\cdot3^n+4\cdot4^n \\ =3\cdot3^n+4^n(3+1)\\
=3\cdot3^n+3\cdot4^n+4^n \\ 
=3(3^n+4^n)+4^n \\
\le 3(5^n)+4^n \\
(5-2)(5^n)+4^n \\
5^{n+1} - 2\cdot5^n+4^n\\
5^{n+1} - 2\cdot(3^n+4^n) +4^n$$
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.

How about this, with the same first step:
$$\begin{align*}
3^{n+1} + 4^{n+1} &\le 3 \cdot 3^n + 4 \cdot 4^n \\
         &\le 5 \cdot 3^n + 5 \cdot 4^n \\
         &\le 5(3^n + 4^n) \\
         &\le 5 \cdot 5^n \\ &= 5^{n+1}
\end{align*}
$$
