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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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    @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?2012-03-30

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How about $a_n = (-1)^n n$ for an oscillating unbounded sequence.

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    @ChibuezeOpata: This is $-1,2,-3,4,-5,6 \ldots $2012-03-29
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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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    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.2014-06-12