Find the infimum of the set {3−n + 5−m} of real numbers, as m and nrange over all positive integers.

To find the infimum of the set {3−n + 5−m}, we need to understand that the infimum is the greatest lower bound of a set.

Step 1: Understand the set The set is composed of elements of the form 3−n + 5−m, where n and m are positive integers. As n and m increase, the value of 3−n and 5−m decrease, because they are in the denominator.

Step 2: Find the lower bound The smallest possible value for each term is when n and m are at their smallest, which is 1. So, the smallest possible value for each term is 3^-1 and 5^-1, which are 1/3 and 1/5 respectively.

Step 3: Add the smallest values The smallest possible value of the sum is therefore 1/3 + 1/5 = 8/15.

Step 4: Check if it's the greatest lower bound Since no other combination of n and m will give a smaller sum, 8/15 is the greatest lower bound, and therefore the infimum of the set.

So, the infimum of the set {3−n + 5−m} is 8/15.