For any X € L = L^1 prove that |EX| ≤ E|X|

For any X € L = L^2 prove that |EX|^2 ≤ E|X|^2

I was trying to prove those facts but I was not sure how to...

I figured out that I need to use the cauchy schwarz inequality but i did not clearly understand how to apply it

I started the questions like this for the first one,

a∙b ≤ |a∙b|≤ |a||b|

I started the questions like this for the second one,

var (X) = E(X^2)-(EX)^2

but this is how far I came, can someone plz help me?