real analysis - $lim_{prightarrowinfty}||x||_p = ||x||_infty$ given $||x||_infty = max(|x_1|,|x_2|)$

I have seen the proof done different ways, but none using the norm definitions provided.

Given:
$||x||_p = (|x_1|^p+|x_2|^p)^{1/p}$ and $||x||_\infty = max(|x_1|,|x_2|)$

Prove:
$\lim_{p\rightarrow\infty}\|x\|_p = \|x\|_\infty$

I have looked at the similar questions:
The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. and Limit of $\|x\|_p$ as $p\rightarrow\infty$ but they both seem to use quite different approaches (we have not covered homogeneity so that is out of the question, and the other uses a different definition for the infity norm).