# Example of a set which is not bounded?

5,667

## Solution 1

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

## Solution 2

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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### Chibueze Opata

Updated on December 06, 2022

• Chibueze Opata 6 months

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.

• The subset of rational numbers (the real numbers are not bounded)
• Really? I don't think so.
• If you think the rationals or the naturals are bounded, then give us an upper bound.
• The way I learnt it is that it's bounded above by + infinity.
• Yes, of course you can do that. But what you are doing is just taking your own definition of "bounded", which does not agree with the common use. Which in particular means that wherever you see the word "bounded" used by someone else, it doesn't mean what you think it means.
• I already explained it's not my own definition. I guess the definition is misleading but Michael Hardy explained this issue.
• @Chibueze Opata: I wouldn't call the definition misleading, I'd call it pointless - after all, every subset of the reals is bounded above by $+\infty$.
• I'd like to say that into the Professor's face...I'll quote you as well :)
• @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?
• • @ChibuezeOpata: This is $-1,2,-3,4,-5,6 \ldots$
• • @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
• 