Would someone like to help with the following question?
Prove that for $n=1,2,\ldots$
(a) $5\leq (4^n+5^n)^{1/n}\leq 10$ and that $(4^n+5^n)^{1/n}$ is bounded,
(b) $(4^n+5^n)^{1/n}\geq (4^{n+1}+5^{n+1})^{1/(n+1)}$,
(c) Hence find $\lim\limits_{n\to\infty} (4^n+5^n)^{1/n}$.
For part (a): Done.
For part (b): I have tried various methods, but am still stuck.
For part (c): (a)+(b) tells us that the given function is decreasing as n gets large, but will never become less than 5. I.e. it converges to a limit NOT LESS THAN 5. Now, I know by taking $\ln$ and then applying L'Hopital's Rule that the required limit is 5. But how to deduce that the limit is exactly 5 just from (a)+(b) alone?
Thanks in advance for any help.