If $x_n \to x$ then $z_n = \frac{x_1 + \dots +x_n}{n} \to x$

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):

In a normed vector space, if $x_n \longrightarrow x$ then $z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x$

I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.

Thanks, bye!

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Divide the sum into two parts; "small indices" and "large indices". For large indices use $|x_i-x|<\epsilon$. For small use $|x_i|\le M$. –  user31373 Jun 8 '12 at 23:43
Check Cesaro mean theorem. –  Mhenni Benghorbal Jun 4 '13 at 1:19

I came up with this (correct me if I'm wrong; I'm new to the forum, and not too sure if I'm supposed to answer to my own question if I'm not 100% sure. I'm 90% sure, though :D )

I know that given $\epsilon >0$ there exists $n_0$ such that if $n\geq n_0$ then $\parallel x_n -x\parallel < \epsilon$

so

\begin{align*} 0 & \leq \left\lVert \frac{x_1 +\cdots +x_n}{n} -x \right\rVert \leq \left\lVert \frac{x_1 + \dots + x_n - nx }{n} \right\rVert \\ & \leq \frac{\lVert x_1 - x \rVert}{n} + \dots + \frac{\lVert x_{n_0 - 1} - x \rVert}{n} + \frac{\lVert x_{n_0} - x \rVert}{n} +\dots + \frac{\lVert x_{n} - x \rVert}{n} \end{align*}

the first $n_{0}-1$ terms are $\leq M$ and the rest are $<\epsilon$ so the expression tends to zero, or I can make it as small as I want it.

Thanks a lot @Leonid Kovalev for the inspiration, though my main problem was that I wasn't aware of what to do with the $nx$ (the silliest part :P)

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This sort of thing is encouraged, I think. –  Dylan Moreland Jun 11 '12 at 3:21
I think you want "$-nx$" in the first line of the display, and $\|x_n - x\|$ at the end. I don't think you want to say that the first few terms are $\leq M$; that doesn't seem to be enough. –  Dylan Moreland Jun 11 '12 at 3:41
@DylanMoreland: Why isn't it enough? Can you explain? –  Bouvet Island Jun 11 '12 at 14:56
@Inti: I think you are missing something in the argument. Generally the argument consists of two steps: first choose $n_0$ such that if $n \geq n_0$, $\|x_n -x \| < \epsilon / 2$. Next choose $N \geq n_0$ such that $M$ (which is at least $\sup_n \|x_n\|$) satisfies $M / N < \epsilon / 2$. I don't see the second part of the argument implemented in your answer. –  Willie Wong Jun 11 '12 at 14:59

There is a slightly more general claim:

PROP Let $\langle a_n\rangle$ be a sequence of real numbers, and define $\langle \sigma_n\rangle$ by $$\sigma_n=\frac 1 n\sum_{k=1}^n a_k$$

Then $$\liminf_{n\to\infty}a_n\leq \liminf_{n\to\infty}\sigma_n \;(\;\leq\;)\;\limsup_{n\to\infty}\sigma_n\leq \limsup_{n\to\infty}a_n$$

P We prove the leftmost inequality. Let $\ell =\liminf_{n\to\infty}a_n$, and choose $\alpha <\ell$. By definition, there exists $N$ such that $$\alpha <a_{N+k}$$ for any $k=0,1,2,\ldots$ If $m>0$, then $$m\alpha <\sum_{k=1}^m \alpha_{N+k}$$

which is $$m\alpha<\sum_{k=N+1}^{N+m}a_k$$

$$(m+N)\alpha+\sum_{k=1}^{N}a_k<\sum_{k=1}^{N+m}a_k+N\alpha$$

which gives

$$\alpha+\frac{1}{m+N}\sum_{k=1}^{N}a_k<\frac{1}{m+N}\sum_{k=1}^{N+m}a_k+\frac{N}{m+N}\alpha$$

Since $N$ is fixed, taking $\liminf\limits_{m\to\infty}$ gives $$\alpha \leq \liminf\limits_{m \to \infty } \frac{1}{m}\sum\limits_{k = 1}^m {{a_k}}$$ (note that $N+m$ is just a shift, which doesn't alter the value of the $\liminf^{(1)}$). Thus, for each $\alpha <\ell$, $$\alpha \leq \liminf\limits_{m \to \infty } \frac{1}{m}\sum\limits_{k = 1}^m {{a_k}}$$ which means that $$\liminf_{n\to\infty}a_n\leq \liminf_{n\to\infty}\sigma_n$$ The rightmost inequality is proven in a completely analogous manner. $\blacktriangle$.

$(1)$: Note however, this is not true for "non shift" subsequences, for example $$\limsup_{n\to\infty}(-1)^n=1$$ but $$\limsup_{n\to\infty}(-1)^{2n+1}=-1$$

COR If $\lim a_n$ exists and equals $\ell$, so does $\lim \sigma_n$, and it also equals $\ell$. The converse is not true.

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