If a set is Pseudo-Cauchy, then it is NOT necessarily Cauchy

Question

The following is the definition of a pseudo-Cauchy sequence:

A sequence $a_n$ is pseudo Cauchy if $\forall$ $\epsilon > 0$, $\exists \ N \in \mathbb N$ such that whenever $n \ge N$, $|a_{n+1} - a_n| \lt \epsilon$.

 

The following is the definition of Cauchy:

A sequence $a_n$ is Cauchy if $\forall$ $\epsilon > 0$, $\exists \ N \in \mathbb N$ such that whenever $m,n \ge N$, $|a_m - a_n| \lt \epsilon$.

I am sure without conviction that not all Pseudo-Cauchy sequences are Cauchy. I can argue this with words, but not symbolically. How can I manipulate this symbolically? My argument is as follows: Cauchy is a more liberal definition relative to Pseudo-Cauchy in the sense that any $m,n$can experience a difference $<\epsilon$. However, Pseudo-Cauchy is strict in the sense that not any $m$ will do. Only consecutive values $n+1,n$ will work. All Cauchy are Pseudo-Cauchy because the $m$'s can be defined to be $m=n+1$. However, it is impossible to reconstruct any $m \ge N$ from $n+1$.

Answer 1

Well it is an act of desperation to ask for and receive the same example repeated in two separate pleas for help. So let's dissect what is happening.

The exercise presumes that the student has already mastered the idea of a Cauchy sequence and is now being tested on whether a weaker version is equivalent. So there are some prerequisites to anyone solving this problem.

You must know (i) what is a Cauchy sequence (ii) the proof that every convergent sequence is Cauchy (iii) the proof that every convergent sequence is bounded (iv) the proof that every Cauchy sequence is bounded (v) the proof that every Cauchy sequence is convergent and (vi) you must have a good solid background in sequence and series limits (in particular you remember the harmonic series as divergent and you have studied rearrangements of conditionally convergent series).

With that background the problem is not troublesome. Without all of that --formidable! Moral: time for a serious review.

Now here is a series of exercises to test your understanding. We define a pseudo-Cauchy sequence as Gooshi has.

Definition Let us say that $\{a_n\}$ is a pseudo-Cauchy sequence if > for every $\epsilon>0$ there is an integer $N$ so that for all $n\geq N$ > the inequality $$|a_n - a_{n+1}|< \epsilon$$ holds.

Ex 1. Show that the sequence $a_n=\sum_{k=1}^n 1/k$ is pseudo-Cauchy but not bounded. [Conclude that it is not convergent.]

Ex 2. Why does this example prove that a pseudo-Cauchy sequence need not be Cauchy?

Ex 3. Find an example of a bounded pseudo-Cauchy sequence that is not convergent. [Hint: same example but use lots of negative terms too.]

Ex 4. Why does this example prove that a bounded pseudo-Cauchy sequence need not be Cauchy?

Try for another attempt at a stronger definition that might be equivalent to Cauchy:

Definition Let us say that $\{a_n\}$ is a Gooshi-sequence if for every $\epsilon>0$ there is an integer $N$ so that for all $n\geq N$ the inequality $$|a_n - a_{n+k}|< \epsilon$$ holds for infinitely many values of $k$.

Ex. 5 Can you find an example of an unbounded Gooshi-sequence?

Ex 6. Can you find an example of a bounded Gooshi-sequence that does not converge?

Answer 2

Here's the canonical example:

Let $a_n =\sum_{k=1}^n \frac1{k} $.

Then $|a_{n+1}-a_n| =\frac1{n+1} \to 0 $ as $n \to \infty $, but $a_n \to \infty$ as $n \to \infty$.