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I would like to verify the validity of the following argument.

Let $f: \mathbb{R}^n/\left\{x_0\right\} \rightarrow \mathbb{R}^+$, $g: \mathbb{R}^n \rightarrow \mathbb{R}^+$ such that $0 \le f(x) \le g(x), \forall x \neq x_0$. Suppose that $\lim_{x \rightarrow x_0} g(x)=0$. I want to show that $\lim_{x \rightarrow x_0} f(x)=0$. Let $\epsilon >0$ be given. Then there exists $\delta>0$ such that for any $x \in \mathbb{R}^n$ such that $0<||x||_2<\delta$ we have that $g(x)<\epsilon$. Hence for any $x: 0<||x||_2<\delta$ we have that $f(x) < \epsilon$ and this proves that $\lim_{x \rightarrow x_0} f(x)=0$.

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Everything is okay except that $\|x\|_{2}$ should be replaced by $\|x-x_0\|_{2}$. –  sos440 Nov 15 '12 at 1:03
    
Oh great, thanks. –  Manos Nov 15 '12 at 1:05
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Also you can note that the positivity of both $f$ and $g$, though not explicitly mentioned throughout the proof, plays a crucial role by serving as a lower bound for the squeezing argument. –  sos440 Nov 15 '12 at 1:09
    
Yes, very good comment, thanks. –  Manos Nov 15 '12 at 1:12

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