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Which of the following is the unbounded set ?

  1. $ X = \{ x\mid x = \frac{1}{n},n \in \mathbb{N} \} $

  2. $ Y = \{ x\mid x = \frac{1}{2^n},n \in \mathbb{N} \} $

  3. $ Z = \{ x\mid x = 2^n,n \in \mathbb{N} \} $

  4. $ W = \{ x\mid x \in \mathbb{N}, x \lt 4532 \} $

After reading the definition from Wikipedia , I can only convince myself that it can't be W but am not getting which is correct from the other three ?

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    @Moron: No they haven't given the definition up-to what I have finished till today.2010-11-10

2 Answers 2

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HINT: When do you say that a set is bounded.

  • A set $X \subset \mathbb{R}$ is said to be bounded if you can find a $M \in \mathbb{R}$ such that $|x| \leq M$ for all $x \in X$.

Now consider your cases:

  • $X = \{ x | x = \frac{1}{n} ; \ n \in \mathbb{N}\}$ *NOTE:* $\frac{1}{n} \leq 1$ for all $n \in \mathbb{N}$

  • Similarly the Second one.

  • Now , the third one is unbounded. (Why?). Assume $2^{n} \leq M$ for all $n \in \mathbb{N}$ and obtain a contradiction.

  • Fourth one follows from the definition.

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    @Chandru1 I think you're missing an absolute value in your definition of a bounded set, or maybe you're just interested in saying it is bounded above.2010-11-10
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A set in $\mathbb{R}$ is bounded if you can find two numbers, $a$ and $b$ such that all elements of the set are between $a$ and $b$. If you want to claim a set is unbounded, whatever $a$ and $b$ I give you, you should be able to find at least one element outside the interval $(a,b)$. Which of your sets satisfies that?

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    Thanks a lot for explaining the concept :)2010-11-10