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## Homework Statement

Let ##X## and ##Y## be nonempty subsets of real numbers such that ##X \subseteq Y## and ##Y## is bounded above. Prove that ##\sup X \le \sup Y##

## Homework Equations

## The Attempt at a Solution

Case 1: ##X = Y##. Trivially, ##\sup X \le \sup Y##.

Case 2: ##X \subset Y##. Then there exists a ##y^* \in Y## that is an upper bound for ##X##. Since ##\sup X## is the least upper bound, ##\sup X \le y^*##. But ##y^* \le \sup Y## since ##\sup Y## is an upper bound for ##Y##. So ##\sup X \le y^* \le \sup Y##, and so ##\sup X \le \sup Y##