the only real difference in showing these things in 1D vs ND is what is used as a measure of length.
In 1D it's absolute value
in 2D it's the vector length, i.e. ||(x,y)|| < delta
using the usual Euclidean metric ||(x,y)|| = sqrt(x 2 + y 2)
so whenever, in your epsilon/delta proof, you need to show that some length is smaller than some number you use this metric rather than just absolute value.
When you go to ND the metric is the same you just include the rest of the vector terms.
See if you can work it now.