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 February 20th, 2009, 08:51 AM #1 Newbie   Joined: Dec 2008 Posts: 4 Thanks: 0 sequence, convergence Suppose that $(x_n)$ is a sequence in $\mathbb{R}$. Define a sequence $(y_n)$ by $y_n=\frac{x_1+x_2+\cdots+x_n}{n}$, $\forall n \in \mathbb{N}$. Prove that if $(x_n)$ converges to $x \in \mathbb{R}$ then $(y_n)$ converges to $x$.
 February 20th, 2009, 09:01 PM #2 Senior Member   Joined: Jul 2008 Posts: 144 Thanks: 0 Re: sequence, convergence to apply Stolz theorem.
 February 25th, 2009, 12:33 PM #3 Senior Member   Joined: Nov 2007 Posts: 258 Thanks: 0 Re: sequence, convergence Find $n_0 : \forall n>n_0, |x_n - x| < \epsilon$. For $n\leq n_0$, set $r_n= x_n-x$. Let $c= r_1 + ... + r_{n_0}$Then, for $n>n_0$, $y_n= \frac{x_1+...+x_{n_0-1}+x_{n_0}+...+x_n}{n} = \frac{(x+r_1)+...+(x+r_{n_0-1})+x_{n_0}+...+x_n}{n} \geq \frac{(x+r_1)+...+(x+r_{n_0-1})+(x-\epsilon)+...+(x-\epsilon)}{n} = x + \frac{c-(n-n_0)\epsilon}{n} \geq x + \frac{c}{n} - \epsilon$ and c/n goes to zero as n goes to infinity. The lower bound is proved similarily.

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