Like in the title:
Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?
Normally this is proved in steps.
The even convergents (i.e. the second, fourth, etc.) are an increasing sequence.
The odd convergents are a decreasing sequence.
All the even convergents are less than all the odd convergents.
The gap between consecutive convergents (one of which is even, and one is odd), decreases as you go further out in the sequence.
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2 years ago