If S and T are bounded, nonempty sets, then sup(S U T) = max{sup(S), sup(t)}.?

Question

Prove that if S and T are bounded, nonempty sets, then sup(S U T) = max{sup(S), sup(t)}.

I start with let x be an element of S U T... where to go from here!?

Answer 1

... then xES and thus x<=sup(S)<=max{sup(S), sup(T)}, OR
xET and thus x<=sup(T)<=max{sup(S), sup(T)}.
Therefore max{sup(S), sup(T)} is an upper bound for S U T.

You must also prove that if y is an upper bound for S U T, then y >= max{sup(S), sup(T)} and therefore max{sup(S), sup(T)} is the least upper bound of S U T.
Proof: xES --> x E S U T --> y>=x therefore y is an upper bound of S and y>=sup(S).
Similarly y>sup(T).
Hence y>=max{sup(S), sup(T)}