Approximating irrational numbers using intervals

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

[-Infinity Infinity]
[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.