Converging sequence of non bounded variation

Question

I can't think of an example of a real sequence, $(x_n)_n$, which converges, but has no bounded variation.

Just to be clear, $(x_n)_n$ has bounded variation if the sequence $(v_n)_n$, defined by $v_n = \sum_{i=1}^n|x_{i+1}-x_i|$, is bounded.

Can someone help me find an example? Or is it the case that all converging sequences have bounded variation?

Answer 1

I explained the idea in a comment but here is a specific example that might be easier to follow.

Let $x_{2n}=\frac1n$ and $x_{2n+1}=0$, $n\ge1$. Then $x_n\to0$ but $|x_{2n+1}-x_{2n}| =\frac1n$ so the variation of this sequence (which I would define as $\sup_{n\ge1} v_n$) is at least as bug as the sum of the harmonic series, which is $\infty$. (That is, if $v_n = \sum_{i=1}^n|x_{i+1}-x_i|$ as in your question, then $v_{2n}$ is at least as big as the $n$-th partial sum of the harmonic series, and we use that the sequence of the partial sums of the harmonic series is unbounded.)