How to prove divergence elementarily

Question

I have this problem: give an example of a real sequence $\;\{a_n\}\;$ with

$$\lim_{n\to\infty}\left(a_{n+1}-a_n\right)=0\;,\;\;\text{but}\;\;\lim_{n\to\infty}a_n\;\;\text{doesn't exist finitely}$$

The two classical examples I know, namely the harmonic series's partial sums sequence, and the sequence $\;\{\log n\}\;$ , both use either infinite series theory or the continuity of the function $\;\log x\;$ , needed to deduce the existence and value of the limit

$$\log(n+1)-\log n=\log\frac{n+1}n\xrightarrow[n\to\infty]{}\log1=0$$

I'm trying to help somebody who has only studied sequences and their limits and is now in the Cauchy sequences part, but has not studied series nor functions and continuity.

Any example in this will be appreciated.

Answer 1

You can also consider the sequence $(a_n)$ which is given by $a_n = \sqrt{n}$.

Note that everything you want to have can be shown elementarily. If $(a_n)$ would converge, then $(a_n^2)$ would also converge, but that is not possible. One can use a similar argument for the unboundedness of $(a_n)$. The convergence of $\sqrt{n+1} - \sqrt{n} \to 0$ can be shown easily, too.

Answer 2

How about partial sums of $$1+\frac12+\frac12+\frac13+\frac13+\frac13+\frac14+\frac14+\frac14+\frac14+\cdots$$

Answer 3

Similar to the other example, how about $$ \frac{1}{2}, \frac{1}{2}+\frac{1}{2}, \frac{1}{2}+\frac{1}{2}+\frac{1}{4},\cdots $$