n this problem for b > a are both real numbers, (a, b] = {c ∈ R : b ≥c > a}. Determine the following subset of R : ∩i∈N (1 − in , 1 + in ]

The problem is asking for the intersection over all natural numbers (i∈N) of the intervals (1 - in, 1 + in].

Step 1: Understand the notation The notation (a, b] represents an open-closed interval, which includes all real numbers between a and b, where a is not included and b is included.

Step 2: Understand the problem The problem is asking for the intersection of all such intervals as i ranges over all natural numbers.

Step 3: Find the intersection The intersection of intervals is the set of all points that these intervals have in common.

Step 4: Analyze the intervals As i increases, the interval (1 - in, 1 + in] becomes wider. However, the lower limit of the interval, 1 - in, decreases as i increases.

Step 5: Find the common points The common points of all these intervals would be the points that are greater than the smallest lower limit and less than or equal to the smallest upper limit.

Step 6: Determine the subset Since the lower limit decreases as i increases, the smallest lower limit would be negative infinity. The smallest upper limit would be 1, which is the upper limit of the interval when i=1.

Therefore, the subset of R that is the intersection of all these intervals is (-∞, 1].