I need to find a sequence of partitions , let's call it S of R=[0,1]x[0,1] such that as the number of partitions k→∞ , then limit of the area of the largest subinterval of the rectangle in the partition, denoted a(S) tends to 0, but the mesh size m(S) is a non-zero value.
The Attempt at a Solution
If I denote Δx_i as the partition width for the i-th interval and say Δx_i = (i2/k2)-((i-1)2/k2) = (2i-1)/(k2). Then the width of the largest subinterval will approach 0 as k→∞, which in turn means the the area of the largest subinterval will go to 0. However, I'm unsure how to show the mesh size for the partition can't be 0? Any help is appreciated.