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Zondrina

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## Homework Statement

Prove or disapprove, for non-empty, bounded sets S and T in ℝ :

sup(SUT) = max{sup(S), sup(T)}

## Homework Equations

The least upper bound axiom of course.

## The Attempt at a Solution

Since we know S and T are non-empty and bounded in the reals, each of them contains a supremum by the least upper bound axiom. Let : L

_{1}= sup(S) ^ L

_{2}= sup(T) be these least upper bounds for S and T respectively.

Since SUT is also a bounded non-empty set, it also contains a supremum by the axiom. Let L = sup(SUT) denote SUT's least upper bound.

We want to show that L = max{L

_{1}, L

_{2}}

Not quite sure how to proceed from here.