How to Find a Sequence of Partitions in Vector Calculus?

[Nicolaus]

Hi, I need help getting a start on this exercise.

Let R be the unit square.
Find sequence of partitions such that the mesh size goes to zero as sequence goes to infinity -> for this, is this just a series of sub rectangles whose respective areas shrink at the same rate with respect to one another?

Find a sequence of partitions such that the max area of a sub rectangle tends to 0 while the mesh size of partition does not tend to zero as the limit approaches infinity of sequence of partitions.

 

[jbunniii]

How are you defining "mesh size"? I'm guessing it is the maximum side length of any of the rectangles in the partition? If so, consider dividing the unit square into $N$ rectangles of equal size, with length $1$ and width $1/N$. Then the mesh size is $1$ regardless of how many rectangles you use.

 

[Nicolaus]

The mesh size is defined as the Euclidean normal of the sub rectangle