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## Homework Statement

Let S be a nonempty subset of R that is bounded below. Prove that inf S = - sup {-s: s in S}

## The Attempt at a Solution

Let a0 = inf S. Thus, for all s in S, a0 is less or equal to s; or -a0 greater or equal to -s.

If u is any upper bound for -S, u is greater or equal to -a0; or -u less or equal to a0.

and here I have difficulty proceeding further. I don't quite see the logic anymore :-(

Any help is greatly appreciated.