(The Principle of Mathematical Induction) For each positive integer

, let

be a statement. If

(1)

is true and

(2) the implication If

, then

is true for ever positive integer

, then

is true for every positive integer

.

*Proof:*Assume, to the contrary, that the theorem is false. Then conditions (1) and (2) are satisfied but there exist some positive integers

for which

is a false statement. Let

is false

Since

is a nonempty subset of

, it follows by the Well-Ordering Principle that

contains

. Since

is true,

.

Here I am lost:
Thus

and

. Therefore,

and so

is a true statement. By condition (2),

is also true

and so

. This however, contradicts our assumption that

.

**Question: What is ?**