Does there exist a sequence $\{a_{n}\}$, but $\lim_{n\to\infty}a_{n}\neq 0$

Question

Does there exist a sequence $\{a_{n}\}$, such that $$\lim_{n\to\infty}(a_{n+1}-a_{n})=0\ \ \ \text{and} \ \ \lim_{n\to\infty}\dfrac{a_{1}+a_{2}+\cdots+a_{n}}{n}=0$$ but $$\lim_{n\to\infty}a_{n}\neq 0 \ ?$$

Answer 1

Assuming that $\lim_{n \to \infty} a_n \neq 0$ means that $a_n$ converges to $a \neq 0$, it does not. Since is quite standard to prove that if $a_n \to a$ then $$ \frac{a_1 + \cdots + a_n}{n} \to a $$ Thus, supposing that $a_n \to a \neq 0$, then $$ \frac{a_1 + \cdots + a_n}{n} \to a \neq 0 $$

Answer 2

This should work for a sequence satisfying the first two hypotheses and $\lim a_n $ failing to exist: Let $T_n$ be the finite sequence

$$\frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1, \frac{n-1}{n}, \dots , \frac{2}{n}, \frac{1}{n}.$$

Let $Z_n$ be the finite sequence of $2^n$ zeros. Then form the sequence

$$T_1\, Z_1 \,T_2\, Z_2 \,\dots \, T_n\, Z_n\, \dots,$$

meaning the finite sequence $T_1,$ followed by the finite sequence $Z_1,$ etc.