# Thread: Limit definition in Rn

1. ## Limit definition in Rn

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

2. ## Re: Limit definition in Rn

You could write down the definition of the left hand limit, and then define a scalar function $g: \mathbb R_n \mapsto \mathbb R$ defined by $g(\mathbf{x}) = \big| |\mathbf{f}(\mathbf{x}) - \mathbf{l}| \big|$ and show that the $(\epsilon, \delta)$ definition of the limit
$\lim_{\mathbf{x} \to \mathbf{c}} g(\mathbf{x}) = 0$

is satisfied.

I think you can do the same thing in reverse to get the other half of the proof.