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.
You could write down the definition of the left hand limit, and then define a scalar function defined by and show that the definition of the limit
I think you can do the same thing in reverse to get the other half of the proof.