Let A and B be a  bounded set

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Let A and B be a  bounded set.

sup(A union B)=max{sup A, sup  B)

I am struggling with the proof

Let x belongs to A to B, then x belongs to A or x belongs to B

x<= sup A, x<= sup B

It means x<=max(sup A, sup B)

i am struggling with the next step

when x<=max(sup A, sup B) implies sup(AUB)<=max (sup A, sup B)

I know that x<=sup(AUB), but I don't know why sup (A U B )<= max (Sup A, sup  B)