Urgent Question: Convergens of series

Could somebody here please verify my proof?

Given the two sequences $\sum_{n=1}^{\infty} c_{n}$ and $\sum_{n=1}^{\infty} d_{n}$ which is defined as

$d_{n} = \frac{c_1 + c_2 + \ldots + c_n}{n}$

1) show that if $c_{n} \rightarrow 0$ , for $n \rightarrow \infty$ is true. Then $d_{n} \rightarrow 0$ for $n \rightarrow \infty$

2) show that if $c_{n} \rightarrow c$ , for $n \rightarrow \infty$ . Then $d_{n} \rightarrow c$ for $n \rightarrow \infty$

Solution 1)

I choose an $N' \in \mathbb{N}$ which for all m, $m \geq N'$ implies that $|c_m|< \frac{\epsilon}{2}$ .

I then choose a random $n > N'$ there is an N'' such that for all

from which is follows that

$d_{n} = \frac{c_1 + c_2 + \ldots + c_N'}{n} + \frac{c_{N'+1}+ c_2 + \ldots + c_n}{n}$

Then by the triangle inequality

$\begin{array}{cccc}|d_n| &\leq& |\frac{c_1 + c_2 + \ldots + c_N'}{n}| + |\frac{c_{N'+1}+ c_2 + \ldots + c_n}{n}| \\ <br /> |d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \frac{|c_{N'+1}|+ |c_2| + \ldots + |c_n|}{n} \end{array} \\<br /> |d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \frac{(N-N')\frac{\epsilon}{2}}{2} \\ |d_n| &\leq& \frac{|c_1 + c_2 + \ldots + c_N'|}{n} + \epsilon \end{array}\end{array}<br />$

there is an N'' such that for all $n \geq N''$ , thus $|d_n| < \epsilon$

Solution 2)

$y_n:=c_n-c \rightarrow 0 \mathrm{ as }n \rightarrow \infty \mathrm{ and thus } d_n-c=\frac{(c_1-c)+\cdots +(c_n-c)}{n}=\frac{y_1+\cdots +y_n}{n}$
will tend to 0 by part (a).

I have let the two last tex modules stand because I cannot get them to work with "math", "/math"

Best regards