Example of a convergent sequence that does not satisfy the condition

Question

Given a sequence $(a_n)$ and there exists a number $c\in \mathbb{R}$ such that for all $n\in \mathbb{N}$ satisfy the condition :

$|a_2 -a_1| +|a_3-a_2| +... +|a_n-a_{n-1}|<c$.

Are there any examples for a convergent sequence $a_n$ that does not satisfy the above condition?

I proved that for that condition, $(a_n)$ converges by Cauchy, but can't see how to find those examples.

Thank you.

Answer 1

I’m confused by the comment chain, why is the alternating harmonic series not an example?

With $(a_n)$ equal to the alternating harmonic series, $|a_{k+1}-a_k| = |\frac{1}{k + 1} - (-\frac{1}{k})|$ or $|a_{k+1}-a_k| = |-\frac{1}{k+1} - \frac{1}{k}|$. In both cases $|a_{k+1} - a_k| = \frac{1}{k + 1} + \frac{1}{k} > \frac{1}{k+1} + \frac{1}{k+1} = \frac{2}{k+1}$.

As such $\sum_{k=1}^N |a_{k+1} - a_{k}| > 2\sum_{k=1}^N \frac{1}{k+1}$, which diverges as $N \rightarrow \infty$. Since this diverges, we cannot find a $c$ which will make this series satisfy the condition. This is despite the fact that $(-1)^k \frac{1}{k+1} \rightarrow 0$ as $k \rightarrow \infty$.