Are These Two Limit Statements Equivalent?

[Poopsilon]

The title basically says it all, here they are:

$$\lim_{n\rightarrow\infty}a_n=a.$$ And $$\lim_{n\rightarrow\infty}|a_n-a|<\varepsilon\hspace{7 mm} \forall\varepsilon>0$$

 

[micromass]

Hi Poopsilon!

Yes, those two things are equivalent. Furthermore they are both equivalent to the statement

$$\lim_{n\rightarrow +\infty}{|a_n-a|}=0$$

The proof is not that hard and follows from the epsilon-delta definitions!

Do you need more explanations?

 

[Poopsilon]

Hey there, this curiosity actually came up rather organically; while trying to prove one statement it led me to consider this more general conjecture:

$$if\hspace{4mm} a_n\rightarrow0\hspace{4mm} as\hspace{4mm} n\rightarrow\infty\hspace{4mm} than \hspace{4mm}\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{a_k}{n}=0.$$

Basically what I did was let epsilon > 0, than use the definition for the convergence of a_n to say that there exists an N such that for all n > N, |a_n| < epsilon. Than from there I split up the series and then used the fact that convergent sequences are bounded and that |a_n| < epsilon for all n > N to bound it by this:

$$\lim_{n\rightarrow\infty}([\sum_{k=1}^{N}\frac{M}{n}]+[\sum_{k=N+1}^{n}\frac{\varepsilon}{n}])$$

which becomes:

$$\lim_{n\rightarrow\infty}(\frac{NM}{n}+\varepsilon\frac{(n-N-1)}{n})=\varepsilon$$

I'm pretty sure this is logically valid, but some how it feels strange because I start with an epsilon-N proof but end by taking a limit on n, makes me a bit uncomfortable. Maybe you can help! =]

 

[micromass]

Seems correct and the proof is quite innovative too!

You might be interested in the following two statements:

1) The Monotone convergence theorem
Consider the double sequence $(a_{m,n})_{m,n}$ such that $a_{m,n}\geq 0$ for every term and such that for every m, the sequence $(a_{m,n})_n$ is monotonically increasing. Then
$$\lim_{n\rightarrow +\infty}{ \sum_{m=1}^{+\infty}{a_{m,n}}}=\sum_{m=1}^{+\infty}{\lim_{n\rightarrow +\infty}{a_{n,m}}}$$

2) The dominant convergence theorem
Consider the double sequence $(a_{m,n})_{m,n}$ such that for all n holds that

$$|a_{m,n}|\leq b_m$$

and such that $\sum_m{b_m}$ converges, then

$$\lim_{n\rightarrow +\infty}{\sum_{m=1}^{+\infty}{a_{m,n}}}=\sum_{m=1}^{+\infty}{\lim_{n\rightarrow +\infty}{a_{n,m}}}$$

The interesting thing about your conjecture is that it doesn't fall under these two theorems...

 

[micromass]

Also interesting is following slight generalization of your conjecture:

$$If~a_n\rightarrow a~\text{then}~\lim_{n\rightarrow +\infty}{\frac{1}{n}\sum_{k=1}^{n}{a_k}}=a$$

The interesting part is that the limit even exists for sequences that do not converge (because they fluctuate too much). For example, if $a_n=(-1)^n$, then we still have

$$\lim_{n\rightarrow +\infty}{\frac{1}{n}\sum_{k=1}^{+\infty}{a_k}}=0$$

The idea is that fluctuating sequences are being "averaged out" and are made more nice.
This general process that you discovered is called "Cesaro summation". It's a very useful idea!

 

[Poopsilon]

I'm working out of Rudin's PMA and those seemed like powerful theorems so I looked them up in the index thinking maybe they would be in the chapter on sequences and series of functions, since they involve the exchange of limit operations and that's the chapter that discusses uniform convergence. He seems to discuss a weakened version of the DCT in that chapter, but saves both in all their glory for the final chapter on Lebesgue Theory (which I haven't covered).

Neither is presented as cleanly as you have presented them above, I suspect he may have generalized them for proper measure-theoretic digestion..

The Monotone Convergence Theorem as you have presented it above seems like it could be used if we put some extra restrictions on 'my conjecture' (since that's what we've called up until now =P) and changed 'monotonically increasing' to 'monotonically decreasing', but I can't be sure.

Either way speaking of Rudin's PMA, its funny the slight generalization of my conjecture you give, since that is essentially the problem in PMA I was working on before I made MY generalization! Ha! But upon reflection I believe you are of course correct, that my conjecture is essentially just the second half of the solution to the problem, except cleaned up some and distilled down to its essence. And moreover upon referring back to the problem, your statement about it holding for certain non-convergent sequences is exactly part b!

Its cool that it has a name, 'Cesaro summation', Rudin sometimes likes to slip in stuff like that into the problems without mentioning its importance in analysis. Anyways all this stuff is very interesting to me, I'm considering analytic number theory as my specialization upon entry into a PHD program, so this type of hard analysis is right up my ally.

 

[micromass]

I'm working out of Rudin's PMA and those seemed like powerful theorems so I looked them up in the index thinking maybe they would be in the chapter on sequences and series of functions, since they involve the exchange of limit operations and that's the chapter that discusses uniform convergence. He seems to discuss a weakened version of the DCT in that chapter, but saves both in all their glory for the final chapter on Lebesgue Theory (which I haven't covered).

Neither is presented as cleanly as you have presented them above, I suspect he may have generalized them for proper measure-theoretic digestion..

Yes, these two theorems are extremely powerful and important theorems in measure theory. In full generality, these theorem say something about exchanging limit and integral (instead of limit and sum, like we did here). But (in measure theory) the sum is just a special case of an integral, so I stated it in this version.

The Monotone Convergence Theorem as you have presented it above seems like it could be used if we put some extra restrictions on 'my conjecture' (since that's what we've called up until now =P) and changed 'monotonically increasing' to 'monotonically decreasing', but I can't be sure.

Hmm, you can't just change montonically increasing to decreasing So the Monotone Convergence Theorem doesn't apply here...

Either way speaking of Rudin's PMA, its funny the slight generalization of my conjecture you give, since that is essentially the problem I was working on before I made MY generalization! Ha! But upon reflection I believe you are of course correct, that my conjecture is essentially just the second half of the solution to the problem, except cleaned up some and distilled down to its essence. And moreover upon referring back to the problem, your statement about it holding for certain non-convergent sequences is exactly part b!

Hehe Well, that solved part (b) then...

Its cool that it has a name, 'Cesaro summation', Rudin sometimes likes to slip in stuff like that into the problems without mentioning its importance in analysis. Anyways all this stuff is very interesting to me, I'm considering analytic number theory as my specialization upon entry into a PHD program, so this type of hard analysis is right up my ally.

Well, I'm pretty certain you will use Cesaro summation in your analytic number theory