Norm, limit, and max norm

Question

For any $\vec{u} \in \mathbb{R}^2$ prove that $$\lim_{p\to \infty} \|\vec{u}\|_p = \max (|u_1|, |u_2|)$$

Then, when $p\to\infty$ we get $||\vec{u}||_{\infty}$. And we get the max norm, $\|\vec{u}\|_{\infty} = \max \sum_{i=1}^n|u_i|$

How do I get from here to $\max (|u_1|, |u_2|)$?

Answer 1

Hint: Suppose that $|u_1| \geq |u_2|$. Then $$ \|u\|_p = |u_1| \left[ 1 + \left(\frac{|u_2|}{|u_1|}\right)^p \right]^{1/p} $$ Now, why will this converge to $|u_1| = \max(|u_1|,|u_2|)$?