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Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?

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How about this one: $\frac{0}{0+\frac{0}{0+\frac{0}{0+...}}}$ :) –  Ali Jun 1 '13 at 16:24

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up vote 5 down vote accepted

Normally this is proved in steps.

  1. The even convergents (i.e. the second, fourth, etc.) are an increasing sequence.

  2. The odd convergents are a decreasing sequence.

  3. All the even convergents are less than all the odd convergents.

  4. The gap between consecutive convergents (one of which is even, and one is odd), decreases as you go further out in the sequence.

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