# Find the flaw in the given proof: about the limit of a sequence

I'm studying Real Analysis, and one problem gives me a trouble. The problem is as below:

Let $$\{x_n\}$$ be a sequence defined on $$\mathbb{R}$$ with $$\displaystyle \lim_{n \to \infty} x_n = x$$ for some $$x \in \mathbb{R}$$. Define a sequence $$\{\sigma_n \}$$ on $$\mathbb{R}$$ by $$\sigma_n= \frac{1}{n} ( x_1 + x_2 + x_3 + \cdots + x_n)$$ Find the flaw of the proof below, which tries to show the claim.

Claim : The sequence $$\{\sigma_n\}$$ converges. In addition, $$\displaystyle \lim_{n \to \infty} \sigma_n = x$$.

Proof. Since $$\displaystyle \lim_{n \to \infty} x_n = x$$, for any $$\epsilon >0$$, there exists a natural number $$N$$ such that $$n >N \quad \Rightarrow \quad \lvert x_n-x \rvert < \epsilon$$ Now fix $$\epsilon >0$$, and let $$N_\epsilon$$ be the natural number that satisfies the property above. Note that $$\lvert \sigma_n -x \rvert = \lvert\frac{1}{n} (x_1 + x_2 + \cdots + x_n) - x\rvert \leq \frac{1}{n} (\lvert{x_1-x}\rvert + \cdots + \lvert{x_n-x}\rvert)$$ Now, for sufficiently large $$n>N_\epsilon$$, we can divide the term above as $$\lvert \sigma_n -x \rvert = \frac{1}{n}(\lvert{x_1-x}\rvert + \lvert{x_2 - x}\rvert + \cdots + \lvert{x_{N_\epsilon}-x}\rvert)+\frac{1}{n}(\lvert{x_{N_\epsilon+1}-x}\rvert + \cdots + \lvert{x_n-x}\rvert)$$ Since the first term above has only finite constant terms, $$\frac{1}{n}(\lvert{x_1-x}\rvert + \lvert{x_2 - x}\rvert + \cdots + \lvert{x_{N_\epsilon}-x}\rvert) \to 0 \quad \text{as} \quad n \to \infty$$ Now, $$\lvert \sigma_n -x \rvert = \frac{1}{n}(\lvert{x_{N_\epsilon+1}-x}\rvert + \cdots + \lvert{x_n-x}\rvert) < \frac{1}{n} \times \epsilon (n-N_\epsilon) \to \epsilon$$ as $$n \to \infty$$. Therefore $$\displaystyle \lim_{n \to \infty} \sigma_n = x$$.

I understand that there is some problem in the proof, but I cannot clearly explain the answer! I think the problem comes from finding the limit not at once, but calculating the parts first. Could somebody explain this to me plainly?

• $s=x{{{{{}}}}}$? Jun 16, 2019 at 17:07
• I don't know if this is the "error" you are looking for but you say that the limit of the second sequence is "s" but have not defined "s"! The limit of the sequence should be "x" as Lord Shark the Unknown suggested. Jun 16, 2019 at 17:12
• Oh, that’s my typo, sorry. The limit of given should also be x! Jun 16, 2019 at 17:14
• Would you mind fixing the parenthesis on the first $|\sigma_n - x| =$ line ? Jun 16, 2019 at 17:14
• There are so many typos in there! I also edited it. Thanks for the comment. Jun 16, 2019 at 17:28

Now, $$\lvert \sigma_n -x \rvert = \frac{1}{n}(\lvert{x_{N_\epsilon+1}-x}\rvert + \cdots + \lvert{x_n-x}\rvert) < \frac{1}{n} \times \epsilon (n-N_\epsilon) \to \epsilon$$ as $$n \to \infty$$.
because you lost the initial $$\frac{1}{n}(\lvert{x_1-x}\rvert + \lvert{x_2 - x}\rvert + \cdots + \lvert{x_{N_\epsilon}-x}\rvert)\neq 0$$ in $$\lvert \sigma_n -x \rvert = \frac{1}{n}(\lvert{x_1-x}\rvert + \lvert{x_2 - x}\rvert + \cdots + \lvert{x_{N_\epsilon}-x}\rvert)+\frac{1}{n}(\lvert{x_{N_\epsilon+1}-x}\rvert + \cdots + \lvert{x_n-x}\rvert)$$
• You mean $\lvert\sigma_n-x\rvert\leq \frac{C(\varepsilon)}{n}+\varepsilon(1-\frac{N}n)$? That is fine. Jun 16, 2019 at 17:45