Prove. Cantor’s principle. For every nested system of intervals [a_n, b_n], n = 1, 2, 3, ... there exists one and only one real number common to all the intervals.

Proof. By the definition of nested intervals,

and

Thus , and the sequences {a_n} and {b_n} are bounded and respectively monotonic increasing and decreasing sequences. Consequently both converge to values that we will call a and b. We now wish to show that a = b.

We start by writing the following identity:

1)        b - a = (b - b_n) + (b_n - a_n) + (a_n - a)

Taking the absolute value of both sides we get

2)        |b - a| |(b - b_n)| + |(b_n - a_n)| + |(a_n - a)|

Now given any ε > 0, we can find an n₀ such that for all n > n₀

Substituting ε /3 for |(b - b_n)|, |(b_n - a_n)| and |(a_n - a)| in 2) we get the inequality

4)        |b - a| < ε

Since ε is any positive number, necessarily b - a = 0 and a = b.

                                                Source: Spiegel. Real Variables. p. 21