I have come across an exercise sheet for my analysis class with the following instructions

In each of the cases below, either (i) give an example with the properties indicated or (ii) show that there is no example that has the properties indicated. Support each answer with a clear and concise mathematical argument.

One of the cases is

A finite set $S \subseteq \mathbb R $ which is unbounded.

Can such a set exist, and if so any examples? Or am I right in presuming an unbounded subset cannot be finite?

  • 7
    $\begingroup$ You are right in presuming that an unbounded set must be infinite. $\endgroup$ May 7 '14 at 13:25
  • $\begingroup$ What does finite mean? What does unbounded mean? Can you write down that to begin with. A clear and concise mathematical argument must is build from definitions and rules, so to begin with, what are these definitions? $\endgroup$
    – String
    May 7 '14 at 13:35

The reals are a totally ordered set. Any subset of a totally ordered set is totally ordered. A finite subset S of the reals is totally ordered and therefore (being finite) has a first and last element f and l with the property that $f \le x \le l$ for all x in S. So S is bounded by f and l.

  • $\begingroup$ That's exactly what I was looking, Clear and concise as to why, and makes perfect sense. Thanks $\endgroup$
    – Paulistic
    May 7 '14 at 16:57

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