1. ## Sequence Limit

Show that if (z_n) = [(a^n + b^n)]^(1/n) where 0 < a < b, then lim(z_n) = b.

I see that b^n will obviously dominate as n goes to infinity, making it (b^n)^(1/n), but how do I show this with algebra or a theorem about sequences (can you use Squeeze Theorem easily?) ?

$\displaystyle \displaystyle \left( a^n + b^n \right)^{1/n} = \left( b^n \left[ \frac{a^n}{b^n} + 1\right] \right)^{1/n} = b \left( \left( \frac{a}{b} \right)^n + 1 \right)^{1/n}$ and remember that $\displaystyle 0 < \frac{a}{b} < 1$.
3. $\displaystyle b^n \leq a^n + b^n \leq b^n + b^n = 2b^n \implies b\leq (a^n+b^n)^{1/n} \leq b\cdot 2^{1/n}$ --> use squeeze theorem.