How to go from limit of vector norm to 'normal' limit

[Berrius]

This is not really a homework question, but I've come across this while preparing for a test

Homework Statement

Let f:U \subseteq R^n -> R^m be a function which is differentiable at a \in U, and u \in R^n
It is then stated that it is clear that:
lim_{t \to 0} \frac{||f(a+t*u)-f(a)-D_f(t*u)||}{||t*u||} = 0 => lim_{t \to 0} \frac{f(a+t*u)-f(a)-D_f(t*u)}{|t|} = 0

How do they get this result?

The Attempt at a Solution

I've tried using the epsilon/delta-def of limits but where the first limit is about real numbers, the second is about vectors in the R^m, so I am totally confused.

 

[mfb]

Denominator: $||t \cdot u|| = |t| \cdot ||u||$ and the limit is linear in $||u||$.
Numerator: The norm of a vector can go to 0 if and only if the vector itself goes to 0.

 

[Berrius]

I know ofcourse ||v||=0 iff v=0, but why is this still true when i put a limit in front of it?

 

[mfb]

You can show it for each component in some basis, for example. They all go to 0.

 

[Vargo]

It is the definition that vector v converges to vector w if the norm of their difference converges to 0.