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## Homework Statement

So I have started my long journey through N.L. Carothers Real Analysis and my intention is to work through every single exercise along the way.

The first problem : http://gyazo.com/ddef0387f04d789c660548c08796585d

## Homework Equations

## The Attempt at a Solution

Suppose A is bounded below, we want to show A has an infimum.

That is, we want to show ##\exists m \in ℝ## such that :

(i) ##m## is a lower bound for A.

(ii) If ##x## is any lower bound for A, then ##x ≤ m##.

So suppose (i) is satisfied ( Since A is bounded below ), we want to show (ii) holds, so we consider the set ##-A = \{ -a \space | \space a \in A \}##.

Since A is bounded below by m, -A is bounded above by -m. Since -A is a nonempty subset of real numbers with an upper bound, ##\exists s \in ℝ \space | \space -m ≤ s##.

Hence sup(-A) = -m ( i.e, -m is the least upper bound of -A ) so that -sup(-A) = m ( i.e, m is the greatest lower bound of A ), but this is precisely what we desire. Thus -sup(-A) = inf(A) = m as desired.

That was my go at it. Any thoughts to make this better? I feel as if my argument wasn't as strong as it should be near the end.