How Do You Prove the Limit of a Vector Function?

[physicsidiot1]

proof for a vector limit??

Homework Statement

Show that the lim(t-->a)r(t)=b. if and only if for every \epsilon>0 there is a number \delta>0 such that if 0<|t-a|<\delta then |r(t)-b|<\epsilon

This is asking to prove the limit of a vector function.
It seems to me that within the question, it is already answered...I don't know how else to show that limit exists other than with what is said above.

Homework Equations

usual epsilon delta stuff

The Attempt at a Solution

I really don't know

 

[LCKurtz]

Homework Statement

Show that the lim(t-->a)r(t)=b. if and only if for every \epsilon>0 there is a number \delta>0 such that if 0<|t-a|<\delta then |r(t)-b|<\epsilon

This is asking to prove the limit of a vector function.
It seems to me that within the question, it is already answered...I don't know how else to show that limit exists other than with what is said above.

I'm guessing here, but perhaps the above is not what you were given for definition of the limit of a vector function. Maybe your definition was given in terms of limits of components, something like this:

Define R(t) = <x(t),y(t),z(t)>. If

lim_t→ax(t) = b₁ and
lim_t→ay(t) = b₂ and
lim_t→az(t) = b₃

then we say lim_t→aR(t) = b =<b₁,b₂,b₃>

If that is the case, you have something to prove.