Approximating irrational numbers using intervals

Every irrational number can be progressively approximated using rational intervals. Here is a sequence of intervals that approximate $\sqrt{2}$:

[0 2]
[4/3 5/3]
[11/8 10/7]
[24/17 27/19]
[65/46 58/41]
[140/99 157/111]
[379/268 338/239]
[816/577 915/647]
[2209/1562 1970/1393]
[4756/3363 5333/3771]
[12875/9104 11482/8119]
[27720/19601 31083/21979]
[75041/53062 66922/47321]
[161564/114243 181165/128103]
[437371/309268 390050/275807]

This sequence of intervals is produced by the continued fraction representation for $\sqrt{2}$ which is 1,2,2,2,2,2,2,... with an infinite sequence of twos.

Interval analysis

The set of all rational intervals is countable. We can develop analysis in terms of this intervals firstly through interval arithmetic:

(= (add-intervals [1 2] [3 4]) [4 6])
(= (multiply-intervals [1 2] [3 4]) [3 8])

We can express any continuous real function as a function of intervals using its set of extrema. For example, the square of the interval [-2,2] is [0,4] and the square of the interval [2,3] is [4,9].