# Real Numbers

• Jan 20th 2009, 01:17 PM
noles2188
Real Numbers
Mark each true or false and justify.

1) If a nonempty subset of real numbers has an upper bound, then it has a least upper bound.

2) Every nonempty bounded subset of real numbers has a maximum and a minimum.

3) If m is an upper bound for S and m' < m, then m' is not an upper bound for S.

4) For each real number x and each E > 0, there exists an n element of the natural numbers such that nE > x.
• Jan 20th 2009, 02:12 PM
ThePerfectHacker
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Originally Posted by noles2188
1) If a nonempty subset of real numbers has an upper bound, then it has a least upper bound.

How about $\displaystyle (-\infty,0]$?

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2) Every nonempty bounded subset of real numbers has a maximum and a minimum.
How about $\displaystyle (-1,1)$?

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3) If m is an upper bound for S and m' < m, then m' is not an upper bound for S.

How about $\displaystyle (-1,1)$ and $\displaystyle m=1$ and $\displaystyle m'=2$?

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4) For each real number x and each E > 0, there exists an n element of the natural numbers such that nE > x.
Yes