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Suppose we know that $0 < S_1 \leq S_2$ and that $a$ belongs to the set $\mathbb{R}^n$.

How would you show that $B(a,S_1)$ is a subset of $B(a,S_2)$?

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Just recall yourself the definition of $B(a,S)$. – Damian Sobota Mar 22 '13 at 1:14
Are $S_1$ and $S_2$ positive real numbers here? – Ian Coley Mar 22 '13 at 1:16
Where exactly do you stuck? – Stefan Hamcke Mar 22 '13 at 1:17
Yes, S1 and S2 are positive real numbers. – Charlie Brown Mar 23 '13 at 3:19

I would use the definition of $B(a,S1)$ as all points within $S1$ of $a$. If a point is within $S1$, it is within $S2$

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$$x\in B(a,r)\iff |x-a|<r$$

Use the fact that $|x-a|<S_1$ then $|x-a|<S_2$.

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