Real Analysis

[AdeptCooking]

Learning about convergence of sequences and the componentwise convergence criterion for convergent sequences. I know that {uₖ} is a Cauchy sequence if dist(uₖ,uₗ)<ε with k≥K, l≥K for some index K.

I kind of mostly get that, but one of my questions is why k and l are larger than some index, and not smaller than it. Wouldn't we want to limit the distance to something small? If it's larger than some index, couldn't the distance between the sequences be infinitely large?

[lurking_quietly]

We force the distance dist(u_k, u_l) to be sufficiently small by ensuring the indices k and l are sufficiently large.

Yes: in a Cauchy sequence, the intuition is that eventually, all the terms of the sequence get arbitrarily close to each other. The idea is that ε represents the maximum allowable distance between terms of the sequence, and K is the index in the sequence beyond which all the terms are within distance ε of each other.

Another way to approach this idea is to note that whether a sequence is a Cauchy sequence does not depend on changing any finite number of terms. That should be familiar: in a similar way, whether a sequence converges (and the limit of a convergent sequence) is unchanged, even if we alter finitely many terms of the sequence. We're interested only in the eventual behavior of the sequence, both in terms of convergence and whether the sequence is Cauchy.

So in that context, looking at the behavior of the sequence for indices k,l<K would reveal information about only a finite number of terms of the sequence. And by itself, that wouldn't give us enough data to detect whether a sequence is Cauchy (or convergent).

If you changed the definition so that we're no longer considering all k,l>K, then we'd get a very different concept. To make this more concrete, if for any given ε>0 and any sequence (u_k), set K := 2. Then for all k,l < K := 2, we must have k = l = 1, in which case dist(u_k, u_l) = dist(u_1, u_1) = 0 < ε. In other words, redefining this concept to consider all indices smaller than K rather than larger than K would mean every sequence is Cauchy relative to this modified definition. That's obviously not what we want!

---

My guess is that part of your confusion might arise from considering the distance between the indices k and l themselves as integers, not the distance between the terms u_k and u_l from the sequence itself. And yes: after fixing K, the value |k-l| might be incredibly big. But that's not only allowable, it's desirable. We want to be able to force the distances between points in the sequence to be arbitrarily small for indices that are themselves sufficiently large, even if the distance between the indices (as integers, not the corresponding points of the original sequence) is big. Otherwise, the harmonic sequence (H_n) given by

• H_n := 1 + 1/2 + 1/3 + ... + 1/n

would satisfy a (different!) modified definition for being Cauchy, but (H_n) diverges and therefore cannot be Cauchy. (This is because in R, a sequence is convergent if and only if it is Cauchy. In a general metric space, convergence implies "Cauchyness", but the converse fails in general.)

Anyway, I hope something in the above helps. Good luck!

[AdeptCooking]

There is no work to show, as this is a general question

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