Proof check of a real analysis problem which i am almost done with

Question

It would be very helpful if you checked my progress on a problem in real analysis.

Problem : If $(a_n)_{n=1}^{\infty} \rightarrow 0$ (in $\mathbb R$) then $\lim_{n \rightarrow \infty} \frac{a_1+a_2+\cdots +a_n}{n}=0$

Proof : Let us define each partial sum $S_n = \sum_{n=1}^{N} \frac{a_1+\cdots a_n}{n}$. We need to show that the sequence $(S_N)_{N=1}^{\infty}\rightarrow 0$.

But as $(a_n)_{n=m}^{\infty} \rightarrow 0$ hence for small $\epsilon$ $\exists M | \forall t\geq M$ we have $|a_t|\leq \epsilon$. Hence, by the triangle inequality, we have $|\frac{a_M+a_{M+1}+\cdots a_{M+k-1}}{k}|\leq \epsilon$. But we have $|S_{M+k-1}-S_{M-1}| = |\frac{a_1+\cdots +a_{M+k-1}}{M+k-1}-\frac{a_1+\cdots +a_{M-1}}{M-1}| \leq |\frac{a_M+a_{M+1}+\cdots a_{M+k-1}}{M+k-1}|\leq |\frac{a_M+a_{M+1}+\cdots a_{M+k-1}}{k}| $ (suitable choice of $k$) $\leq \epsilon$ implying the sequence described is Cauchy and as Reals are a complete metric space hence we have that it converges. But how to show it converges to zero? I tried to show that $|S_m|\leq \epsilon$ $\forall$ small $\epsilon > 0$ by using triangle inequality but obviously it holds for a $t>1$ usually, so i cannot start the sum with $a$'s index being $1$ exactly each time.

Answer 1

Unfortunately, your approach is incorrect. To begin with, the definition $$S_n = \sum_{n=1}^{N} \frac{a_1+\cdots a_n}{n}$$ doesn't make sense. $n$ is a dummy variable on the right-hand side of the equation; it's just the index fo summation. You can't use it as an unbound variable on the left-hand side. Also you seem to be thinking of a sum of averages, but the question is about a sequence of averages. You mean $$S_N =\frac1N\sum_{n=1}^N a_n$$

It isn't true that $|S_N - S_{N+k}|$ is the average of the last $k$ term, so you haven't proved that the sequence is Cauchy.