how could I prove that $l^p$ is a uniformly convex Banach Space

Question

I am trying to apply Milman-pettis Lemma to show that $l^p$ is reflexive hoping that Milman-pettis Lemma would do the heavy lifting and was hoping to easily prove uniform convexity of $l^p$ It turns out I was wrong . I thought triangle inequlality might work but I was wrong and nothing else seems to be working. Any hints on how could I go about doing this? Is the proof hard? Because its been 3 semesters that I started taking university level Math courses. P.S I do know how to show that the canonical embedding of $l^p$ into $(l^p)^{**}$ is surjective. Please note I am referring to ($1<p<\infty$) Thanks