# If $a_n\to \ell$ then $\hat a_n\to \ell$

I need some help to finish this proof:

THEOREM

Let $\{a_n\}$ be such that $\lim a_n=\ell$ and set

$$\hat a_n=\frac 1 n \sum_{k=1}^na_k$$

Then $\lim\hat a_n=\ell$

PROOF

Let $\epsilon >0$ be given. Since $\lim a_n=\ell$ , there exists an $N$ for which $$\left| {{a_n} – \ell } \right| < {\epsilon/2 }$$

whenever $n>N$. Now:

$$\begin{eqnarray*} \left| {\frac{1}{n}\sum\limits_{k = 1}^n {{a_k}} – \ell } \right| &=& \left| {\frac{1}{n}\sum\limits_{k = 1}^n {\left( {{a_k} – \ell } \right)} } \right| \\ &=& \left| {\frac{1}{n}\sum\limits_{k = 1}^N {\left( {{a_k} – \ell } \right)} + \frac{1}{n}\sum\limits_{k = N + 1}^n {\left( {{a_k} – \ell } \right)} } \right| \\ &\leqslant& \frac{1}{n}\sum\limits_{k = 1}^N {\left| {{a_k} – \ell } \right|} + \frac{1}{n}\sum\limits_{k = N + 1}^n {\left| {{a_k} – \ell } \right|} \\ & <& \frac{N}{n}\zeta – \frac{{N }}{2n}\epsilon+\epsilon/2 \end{eqnarray*}$$

where $$\zeta=\mathop {\max }\limits_{1 \leqslant k \leqslant N} \left| {{a_k} – \ell } \right|$$

Now, let $n_0$ be such that if $n>n_0$,

\eqalign{ & \frac{{N\zeta }}{n} < {\epsilon} \cr & \frac{N}{n} < {1} \cr}
Then we get

$$\frac{N}{n}\zeta – \frac{N}{n}\frac{\epsilon }{2} + \frac{\epsilon }{2} < \epsilon – \frac{\epsilon }{2} + \frac{\epsilon }{2} = \epsilon$$

How far is this OK? Do you think there is an easier way to go about proving it? I now remember that by Stolz Cesàro:

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{k = 1}^n {{a_k}} = \mathop {\lim }\limits_{n \to \infty } \frac{{\sum\limits_{k = 1}^{n + 1} {{a_k}} – \sum\limits_{k = 1}^n {{a_k}} }}{{n + 1 – n}} = \mathop {\lim }\limits_{n \to \infty } {a_{n + 1}} = \ell$$

#### Solutions Collecting From Web of "If $a_n\to \ell$ then $\hat a_n\to \ell$"

For the tail sum
$$\frac1n\sum_{N+1}^n|a_k-\ell|\leq \frac{(n-N)}{n}\varepsilon<\varepsilon$$
for all large $n$, hence
$$\limsup_n \frac1n\sum_{1}^n|a_k-\ell|\leq 0+\varepsilon=\varepsilon…$$

The sequence $\{|a_n-\ell|\}$ converges to $0$ therefore it’s bounded, so there exist some $M\gt 0$ such that
$$|a_n-\ell|\leq M\quad\forall n\in \Bbb N.$$
This $M$ plays the role of the $\zeta$ in your proof, but notice it does not depends on $n$.

Let $\epsilon\gt 0$. There exist $n_1\in\Bbb N$ such that for all $n\in\Bbb N$, $n\geq n_1$ implies $$|a_n-\ell|\lt\frac{\epsilon}{2}.$$

Since $$\lim_{n\to\infty} \frac{n_1}{n}=0,$$ there exist $n_2\in\Bbb N$ such that, for all $n\in\Bbb N$, $n\geq n_2$ implies $$\frac{n_1}{n}\leq \frac{\epsilon}{2M}.$$

Let $N=\max\{n_1,n_2\}$, then for all $n\in\Bbb N$, $n\geq N\geq n_1$ implies
$$\left(1-\frac{n_1}{n}\right)\leq 1$$
and then
\begin{align*} \left| -\ell+\frac1{n}\sum_{j=1}^n a_j \right| &\leq \frac1{n}\sum_{j=1}^n \left|a_j-\ell\right|\\ &= \frac1{n}\sum_{j=1}^{n_1} \left|a_j-\ell\right| + \frac1{n}\sum_{j=n_1+1}^{n} \left|a_j-\ell\right|\\ &\lt \frac1{n} n_1M + \frac1{n}(n-n_1)\frac{\epsilon}{2}\\ &=\frac{n_1}{n}M + \left(1-\frac{n_1}{n}\right)\frac{\epsilon}{2}\\ &\lt \frac{\epsilon}{2M}M + 1\cdot \frac{\epsilon}{2}=\epsilon. \end{align*}

Therefore $$\hat a_n\to\ell.$$