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I have a question.

I have to show that the following set is bounded: $$V = \{x \in \mathbb{R}^n : |x_i|≤ \alpha_i,i = 1,...,n\} $$

You can see that $|x_i|$ is always bounded, but how to prove this formally?

Thank you

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    The given definition of $V$ seems enough to say that it is bounded. So how else do you define boundedness ? – 2017-01-30

1 Answers 1

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You can easily see that the set is contained in $$\overline{B^n(0, \alpha)}$$ where $\alpha=\sqrt{n}\max\{\alpha_1,\alpha_2,\dots, \alpha_n\}$

Here, $B^n(x_0, r)$ denotes a ball centered around $x$ with a radius of $r$, i.e.

$$B^n(x_0,r)=\{x\in\mathbb R^n: \|x-x_0\|

You can show that, for every $x\in V$, you have $$\|x-0\|=\|x\|=\sqrt{|x_1|^2+|x_2|^2+\dots + |x_n|^2}\leq \sqrt n\alpha$$