I have a question.
Let $f: \mathbb{R}^n \to \mathbb{R}^m$ and let $L \in \mathbb{R}^m$. Show that
$\lim_{x \to c}f(x) = L$ if and only if $\lim_{x \to c} \| f(x) - L\| =0$.
How can I proof this?
Thank you
Continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$
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real-analysis
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0Break it into pieces since it is a vector-to-vector mapping. Then you can use triangle inequality to work the limit. – 2017-01-30
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1If you remember the ‘epsilon-delta’ ($\epsilon$-$\delta$) definition of the limit, the result is obvious. Think: how could you prove the same for a one-variable real function? – 2017-01-30