A question on sub-sequential limits of a partial sum of a convergent sequence.

Question

Let: $$ \lim_{n\to\infty} x_n = 0 $$ Define a sequence $\{S_n\}$: $$ S_n = x_1 + x_2 + \cdots +x_n\\ n\in\Bbb N $$ Is it possible for $S_n$ to have only two sub-sequential limits if:

1. $x_n$ is in reals, $x_n \in \Bbb R$
2. $x_n$ is in complex numbers, $x_n \in\Bbb C$

I've started with supposing that $S_n$ has two distinct finite sub-sequential limits, namely: $$ \begin{cases} \lim_{n_k \to \infty} S_{n_k} = a\\ \lim_{n_p \to \infty} S_{n_p} = b\\ a, b \in \Bbb R\\ a\ne b \end{cases} $$

I've shown earlier that: $$ \lim_{n\to\infty} S_n = L \in \Bbb R \implies \lim_{n\to\infty}x_n = 0 $$

So if we use this fact for both subsequences of $S_n$ we may obtain: $$ \lim_{n_k \to \infty} S_{n_k} = a \implies \lim_{n_k \to\infty} x_{n_k} = 0 \\ \lim_{n_p \to \infty} S_{n_p} = b \implies \lim_{n_p \to\infty} x_{n_p} = 0 $$

By initial conditions we are given that: $$ \lim_{n\to\infty}x_n = 0 $$ Thus any subsequence of $x_n$ converges to the same limit, namely: $$ \lim_{n_k \to\infty}x_{n_k} = \lim_{n_p \to\infty}x_{n_p} = \lim_{n \to\infty}x_{n} = 0 $$

So in case two subsequences of $S_n$ converge to some number then it implies $x_n$ converges to $0$. In general it seems like $S_n$ may have infinitely many sub-sequential limits and even in this case $x_n$ will still converge to $0$.

1. My first question is about the correctness of the argument above. Is the above enough to consider the question answered? If not what would be the right way?

2. The second question is how do I handle the complex numbers case? I've never taken any course on complex analysis and only have some basic understanding of the complex plane.

Answer 1

I've shown earlier that: $$\lim_{n \to \infty} S_n = L \in \Bbb R \implies \lim_{n \to\infty} x_n = 0$$ So if we use this fact for both subsequences of $S_n$ we may obtain: $$\lim_{n_k \to \infty} S_{n_k} = a \implies \lim_{n_k \to\infty} x_{n_k} = 0$$

What you had shown earlier can be more generally stated as: For a sequence $\{y_k\}$, $$\lim_{K\to\infty} \sum_{k=1}^K y_k = L \in \Bbb R \implies \lim_{k \to \infty} y_k = 0$$ where I've changed all the names to protect the innocent. I.e., to remind you that the sequences are not necessarily the same as the ones you are talking about in the rest of your post.

Now, if you want to prove $\lim_{n_k \to\infty} x_{n_k} = 0$ with this, then $y_k = x_{n_k}$, so what you need to know is that $\lim_{K\to\infty}\sum_{k=1}^K x_{n_k}$ converges. But you have $\lim_{K\to\infty} S_{n_K} = 0$. Note that $$\sum_{k=1}^K x_{n_k} = x_{n_1} + x_{n_2} + x_{n_3} + ... + x_{n_K}$$ while $$S_{n_k} = x_1 + x_2 + x_3 + ... + x_{n_k}$$ These are not the same. So you need to do some more work here.

Thus any subsequence of $x_n$ converges to the same limit

Not "any". You didn't start out by assuming that $x_{n_k}$ or $x_{n_p}$ were arbitrary subsequences. Even if you correct the earlier argument (and it is a relatively easy fix), all you've shown is that $\lim_k x_{n_k} = 0$ for convergent subsequences of $S_n$.

So in case two subsequences of $S_n$ converge to some number then it implies $x_n$ converges to $0$.

Where did that come from? It doesn't follow logically from what you've said before at all.

A more general problem is that this "proof" is not even headed in a direction you need to go. You went through this and then declared in the end that $\lim_n x_n = 0$. but that is a statement you already knew. Arriving at it means you've gone nowhere.

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What you need to consider is this: If subsequences of $S_n$ converge to both $a$ and $b$ with $b > a$, this means that when you start summing $x_n$, eventually the sum will get close to $a$. As you keep summing, it may drift away from $a$, but eventually, it'll end up coming back to $a$ over and over again, all the way to infinity. But the same can be said about $b$. The sum must keep coming near to it as well. So it has to go near $a$, then go near $b$, then later it must go near $a$ again, and then back to $b$, and so on. It must oscilate back and forth between $a$ and $b$ forever. but since $\lim_n x_n = 0$, it has to keep doing this oscilation in smaller and smaller steps. So what happens if $c$ is between $a$ and $b$?

As for $\Bbb C$, there is nothing in this problem that makes any use of the algebraic properties of $\Bbb C$. As far as you need to be concerned here, $\Bbb C$ is just another name for the plane $\Bbb R^2$. Can you see how a second dimension might offer a way for $S_n$ to get from $a$ to $b$ and back without covering the same country in the middle every time?