a) My immediate take on this (long time since I've done topology) is as follows.
Since A is connected, there exists a continuous path from u to v.
Thus (I expect there's a theorem somewhere that proves this), the projection of this path onto each of the 3 co-ordinates is also continuous.
That is, the x-coordinate of that path must be continuous from 0 to 4, the y-coordinate from 0 to 3 and the z-coordinate from 1 to 0.
As the y-coordinate is continuous from 0 to 3, it must pass through 2 somewhere along it.
b) I'll have to pass on that one as I can't remember what a norm is.