I have a question.
Let f : Rn → Rm and let l ∈ Rm. Show that lim x→c f(x) = l ⇔ lim x→c||f(x)−l|| = 0.
How can I do this? I know the (ε, δ)-definition of limit, but this is in Rn.
Thank you
You could write down the definition of the left hand limit, and then define a scalar function $\displaystyle g: \mathbb R_n \mapsto \mathbb R$ defined by $\displaystyle g(\mathbf{x}) = \big| |\mathbf{f}(\mathbf{x}) - \mathbf{l}| \big|$ and show that the $\displaystyle (\epsilon, \delta)$ definition of the limit
$\displaystyle \lim_{\mathbf{x} \to \mathbf{c}} g(\mathbf{x}) = 0$
is satisfied.
I think you can do the same thing in reverse to get the other half of the proof.