# Convergence of a sequence on a normed space

I'm trying to solve an Analysis problem.

I have a given sequence $$(x_n)_n$$ and $$(y_n)_n$$ defined by $$y_n = 1/n \sum _{ k=1 }^{ n }{ x_{ n } }$$. I have to prove

1. $${ ({ x }_{ n }) }_{ n }converges\quad \Longrightarrow { ({ y }_{ n }) }_{ n }converges$$

I haven't found any counterexamples for it so I tried to prove it but i don't know how to start. Any ideas?

• Just for clarification, you mean that $y_n$ is the arithmetic mean of $x_1, \ldots, x_n$, right? That is, $$y_n = \frac{1}{n} \Sigma_{k=1}^n x_k$$ right? – TM Gallagher Mar 6 at 3:14

If $$x_n$$ converges to $$a$$, then for any $$\varepsilon$$ there is an $$N$$ such that for all $$n>N$$, $$|x_n-a|<\varepsilon$$. Thus if the sum of the first $$N$$ terms is $$s$$, then $$\frac{s+k(a-\varepsilon)}{N+k}\leq y_{N+k}\leq \frac{s+k(a+\varepsilon)}{N+k}$$
Then the $$s/(n+k)$$ term approaches 0, and $$k/(N+k)$$ approaches 1, thus for any $$\varepsilon$$, there is some $$M$$ such that $$a-2\varepsilon, so the limit $$b$$ of $$y_n$$ equals a.
However, the converse is not true: consider $$x_n=(-1)^n$$. Hope this helps!