Proof of the convergence of a sequence Assume f:[a,b] $\Rightarrow \mathbb{R}$ is continuous, f(x) $\geq$ 0 for all x $\in$ [a,b], and M = sup{f(x) : x $\in$ [a,b]}. Show that
$$\left\{\left[\int_a^b [f(x)]^n dx \right] ^{\frac1n}\right\}_{n=1}^\infty$$
converges to M. 
Not only can I not prove this, I can't see intuitively why this would be true. Can someone explain anything about this to me?
Sorry the formatting is poor. While I'm at it, can whoever edits this tell me how to make it so squiggly brackets actually show up in formulas? They also just screw up the whole thing and put a box around it. Ideally there would be big squiggly brackets around everything except the bounds n=1 to infinity.
 A: What a super problem this is.  Let $a_n = (\int_a^b f(x)^{n}\, dx)^{n^{-1}}$ be the expression you wrote in your question. 


*

*Observe that $f \leq M \implies a_n \leq M$. EASY

*We will now show that for each $\epsilon>0$ that there exists some $N(\epsilon)$ such that whenever $n>N(\epsilon)$ we have that $a_n \geq M-\epsilon$. HARDER
Therefore, 1 and 2 together imply that for sufficiently large $n$, we have $M-\epsilon \leq a_n \leq M$.  This proves that $a_n \rightarrow M$.
Let's show 2. Let $\epsilon>0$. Since $f$ is continuous on a compact set, we know two things: a) $f(c) = M$ for some $c \in [0,1]$; and b) there exists some $.5>\delta>0$ such that $x \in (c-\delta,c+\delta) \implies f(x)\geq M-\epsilon$.  
Therefore we have that $(\int_a^b (f^n))^{n^{-1}} \geq (\int_{c-\delta}^{c+\delta} (f^n))^{n^{-1}} \geq (\int_{c-\delta}^{c+\delta} (M-\epsilon)^n)^{n^{-1}} = ((M-\epsilon)^n(2\delta))^{n^{-1}} = (M-\epsilon)(2\delta)^{n^{-1}}$.
Now we have a fact: $(2\delta)^{n^{-1}} \nearrow 1$.
Therefore choose an $N(\epsilon)$ such that $n>N(\epsilon) \implies (2\delta)^{n^{-1}}>\frac{M-2\epsilon}{M-\epsilon}$.  Letting $n>N(\epsilon)$ now gives that $(\int_a^b f^n\, dx)^{n^{-1}} \geq M-2\epsilon$.
As $\epsilon>0$ arbitrary, the claim is proven.
For the record, I have no intuition on this, I just wanted to grab onto the supremum and see what happens.
