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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 n_{0} such that for all n > n_{0}

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

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