analysis PLEASE HELP!

You might like to prove

(1) max{sup(S1),sup(S2)} is an upper bound for S1uS2

(2) if z < max{sup(S1),sup(S2)} then z is not an upper bound for S1uS2

You might like to prove

(1) max{sup(S1),sup(S2)} is an upper bound for S1uS2

(2) if z < max{sup(S1),sup(S2)} then z is not an upper bound for S1uS2

Suppose that x $\in$ S1uS2. Then (i) x $\in$ S1, or (ii) x $\in$ S2 (as you said).

So x <= max{sup(S1), sup(S2)}.

ie, max{sup(S1), sup(S2)} is an upper bound for S1uS2.

You haven't yet stated that max{sup(S₁), sup(S₂)} is in fact the *least* upper bound of S₁US₂...

To finish off the proof:

let y < max{sup(S₁), sup(S₂)}

by aritrary choice:
if max{sup(S₁), sup(S₂)} = sup(S_i)
y < sup(S_i)

suppose y is an upper bound of S₁US₂

there does not exist an element of S₁US₂ > y
however S₁US₂ contains all elements of S_i