Is this sequence Cauchy?

[匿名]

2013-12-24 2:26 pm

something like

0, 1, 1/2, 0, 1/3, 2/3, 1, 3/4, 2/4, 1/4, 0, 1/5, 2/5, 3/5, 4/5, 1, 5/6, 4/6, 3/6, 2/6, 1/6, 0,...

It's Cauchy because the distance between the terms is getting smaller, but it's not convergent because it keeps fluctuating between 0 and 1. And it is true for any epsilon the distance would be close enough because the know the rationals are dense in the reals.

But we know a sequence is Cauchy if and only if it's convergent. But this is not convergent so cannot be Cauchy right?

Thanks.

回答 (1)

2013-12-24 5:57 pm

✔ 最佳答案

A sequence {a(n)} is Cauchy if given any ε > 0, we can find a positive integer N such that
|a(m) - a(n)| < ε for all m, n > N.
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The given sequence is not Cauchy because the distance between elements is not getting arbitrarily small (as governed by ε); in particular both 0 and 1 occur infinitely often in the sequence and are always 1 unit apart. (In particular, |a(m) - a(n)| < ε for all sufficiently large m and n is false for ε < 1 due
to the presence of the 0s and 1s.)

I hope this helps!

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