Prove that the sequence $(x_n)$ is Cauchy.

Question

Can you find errors in my prove?

Problem:

Let $(x_n)$ be a sequence of real numbers such that $(y_n)$ is a limited sequence with general term $y_n = \sum_{k=1}^{n}|x_{k+1} - x_k|$. Prove that $(x_n)$ is Cauchy.

My attempt:

$y_n = \sum_{k=1}^{n}|x_{k+1} - x_k| = |x_{n+1} - x_1| \geq |x_{n+1}| - |x_1| \Rightarrow |x_1| + y_n \geq |x_{n+1}|$, where I use the telescopic property in the second equality. Since $y_n$ is convergent (because it is monotonic and limited), $|x_{n+1}|$ is also convergent by the comparision criterion, therefore $x_n$ converges absolutely, therefore it is Cauchy.

sorry if this is duplicated.

Answer 1

The sum $\sum_{k=1}^n |x_{k+1}-x_k|$ is not telescopic, because of the absolute value ($\sum_{k=1}^n x_{k+1}-x_k$ would be). What is actually true is that $(y_n)_{n\in\mathbb{N}}$ is bounded and monotonic, thus convergent. Then $(y_n)_{n\in\mathbb{N}}$ is Cauchy, and for every $\epsilon\in\mathbb{R}^+$ there exists $N(\epsilon)\in\mathbb{N}$ such that $|y_{n+p}-y_n|<\epsilon$ for all $n\geq N(\epsilon)$, $p\in\mathbb{N}$. But using triangular inequality: $$|x_{n+1+p}-x_{n+1}|\leq \sum_{k=1}^{p} |x_{n+k+1}-x_{n+k}|=\sum_{k=1}^{n+p}|x_{k+1}-x_{k}|-\sum_{k=1}^{n}|x_{k+1}-x_{k}|=y_{n+p}-y_{n}<\epsilon$$ So $(x_n)_{n\in\mathbb{N}}$ is Cauchy sequence.

Answer 2

You've showen $(x_n)$ is a bounded ("limited") sequence, but that's not enough for convergence. There's no comparison test that says $a_n \le b_n$ and $b_n$ converges implies $a_n$ converges.

To prove the statement, use the fact that $y_n$ is a Cauchy sequence (you've shown it converges) to show $x_n$ is a Cauchy sequence directly with a similar triangle inequality application.