- #1

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## Homework Statement

http://gyazo.com/27a104e05c409c75611ee4250a89c790

## Homework Equations

Sup/Inf axioms as well as the ε-N definition.

## The Attempt at a Solution

Suppose A is a nonempty subset of ℝ bounded above by ##M##. We want to show that ##lim_{n→∞} x_n = sup(A)## where ##x_n## is a sequence of elements of A.

That is, ##\forall ε > 0, \exists N \space | \space n ≥ N \Rightarrow |x_n - sup(A)| < ε##

Note that since A is bounded above by M, we know that ##M > a, \space \forall a \in A## including ##sup(A)##.

So :

##|x_n - sup(A)| ≤ |x_n| + |sup(A)| < |x_n| + M##

So ##|x_n|## is bounded above by ##ε - M## and below by ##-(ε+M)##

I'm not quite sure how to continue here.