or real numbers c and d, c < d, recall that (c, d] = {b ∈ R : d ≥b > c} and that (c, d) = {b ∈ R : d > b > c}. Find the set of all realnumbers in ∩u∈(0,1)(−u2, 1]

The intersection ∩u∈(0,1)(−u^2, 1] represents the set of all real numbers that are in every single one of the sets (−u^2, 1] for each u in the interval (0,1).

Step 1: Understand the sets The set (−u^2, 1] includes all real numbers greater than -u^2 and less than or equal to 1. As u ranges from 0 to 1, -u^2 will range from 0 (when u=0) to -1 (when u=1).

Step 2: Find the intersection The intersection of all these sets is the set of all real numbers that are greater than every value of -u^2 for u in (0,1) and less than or equal to 1. The smallest value of -u^2 as u ranges from 0 to 1 is -1. Therefore, the intersection of all these sets is the set of all real numbers greater than -1 and less than or equal to 1.

So, the set of all real numbers in ∩u∈(0,1)(−u^2, 1] is (-1, 1].