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It suddenly occurred to me that the set of real numbers is bounded. So suddenly, I'm wondering: What is an example of a set that is unbounded.

NOTE: This question was triggered when I came across oscillating sequences (which can be finite or infinite), so I was wondering an example of an infinite oscillating sequence.

Thanks.

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    $\begingroup$ The subset of rational numbers (the real numbers are not bounded) $\endgroup$ – M Turgeon Mar 29 '12 at 12:39
  • $\begingroup$ Really? I don't think so. $\endgroup$ – Chibueze Opata Mar 29 '12 at 12:53
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    $\begingroup$ If you think the rationals or the naturals are bounded, then give us an upper bound. $\endgroup$ – Martin Argerami Mar 29 '12 at 12:58
  • $\begingroup$ The way I learnt it is that it's bounded above by + infinity. $\endgroup$ – Chibueze Opata Mar 29 '12 at 13:11
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    $\begingroup$ @ChibuezeOpata: what definition are you talking about? You'll be hard pressed to find a mathematician that says that $\mathbb{N}$ or $\mathbb{R}$ is bounded. The word "bounded" is very common in the mathematical literature, and it does not mean what you say it means. According to you, every function is "bounded", every subset of $\mathbb{R}$ is "bounded"... As Martin Wanvik said above, what's the point of even using the word bounded then? $\endgroup$ – Martin Argerami Mar 30 '12 at 3:47
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How about $a_n = (-1)^n n$ for an oscillating unbounded sequence.

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  • $\begingroup$ :) Can you provide example, say starting from -1 for this set? $\endgroup$ – Chibueze Opata Mar 29 '12 at 12:54
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    $\begingroup$ @ChibuezeOpata: This is $-1,2,-3,4,-5,6 \ldots $ $\endgroup$ – Ross Millikan Mar 29 '12 at 13:05
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Within the set $\mathbb{R}\cup\{\infty\}\cup\{-\infty\}$, the set $\mathbb{R}$ is bounded, but usually what is meant by "bounded" is having upper and lower bounds within $\mathbb{R}$. By that usual definition $\mathbb{R}$ is not bounded. Among other unbounded sets are the set of all natural numbers, the set of all rational numbers, the set of all integers, the set of all Fibonacci numbers. All finite sets are bounded. Many infinite sets are bounded as well---for example, the set of all numbers between $0$ and $1$, and the numbers in the sequence $1,1/2, 1/3, 1/4,\ldots\ {}$.

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  • $\begingroup$ This was really a useful answer. Thanks. $\endgroup$ – Chibueze Opata Mar 29 '12 at 13:20
  • $\begingroup$ @ChibuezeOpata Note that $\mathbb{R} \cup \{\infty, -\infty\}$ is not the set of real numbers. It is usually called the system of "extended reals": en.wikipedia.org/wiki/Extended_real_number_line $\endgroup$ – M. Vinay Jun 12 '14 at 10:37
  • $\begingroup$ Nice, what I had been looking for actually is a defined sequence that can take up to negative and positive infinities... Which is what Matt provided. $\endgroup$ – Chibueze Opata Jun 12 '14 at 10:47

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