>>12071812Okay I don't know if this is what you're struggling with, but it's always important to know how to decode inequalities with inf and sup. In this case you have something like inf M <= inf N, which is equivalent to for all n in N there exists an m in M with m <= n.
So you let be coverings of respectively. The goal is to find a covering of
so that the inequality with the sum of lengths of these intervals holds.
Well if you think about it, the set is gonna be covered at each point at least once, but sometimes even twice (if the point is covered by an A_i and also a B_j). Well if you have any such double covered point, you could shorten the B_j a little so that it still covers the point, but the total sum lf lengths has been reduced!
