Problem 3. Prove: For every positive integer n,
∑_{k=1}^n 1/k" ≤ 2 - 1/n.
The demonstration that all of mathematics can be carried out within the framework of set theory includes the following —definition— of the natural numbers. First, the number 0 is defined to be ∅, the empty set. Next, for each previously defined natural number n, the number n + 1 is defined to be the set n ∪ {n}.
Problem 4. (a) Write out the numbers 1, 2, and 3, defined as above.
(b) Prove: For every n ∈ ℕ, n = {k ∈ ℕ | k < n}.