A sequence $\{x_n\}$ is said to have bounded variation if a sequence $\sigma_n$ is bounded, where $\sigma_n$ is defined as:
$$
\sigma_n = |x_2 – x_1| + |x_3 – x_2| + \cdots + |x_{n+1} – x_n|,\ n\in\Bbb N
$$
Prove that any monotone bounded sequence $\{x_n\}$ has bounded variation.
I've started with taking a look at $\sigma_n$, since it is a sum of absolute values then it must follow that:
$$
\sigma_{n+1} \ge \sigma_n
$$
Thus $\sigma_n$ is monotonically increasing. To show a sequence is bounded it is sufficient to show that it converges.
Here is what we wan't to show eventually:
$$
\exists\lim_{n\to\infty}x_n = L_1 \implies \exists M\in\Bbb R: \sigma_n \le M\ \forall n\in\Bbb N
$$
It is given that $\{x_n\}$ is bounded and monotonic, thus it converges by Monotone Convergence Theorem. Define a new sequence:
$$
y_n = x_{n+1} – x_n
$$
By convergence of $x_n$ it follows that:
$$
\exists \lim_{n\to\infty}y_n = \lim_{n\to\infty}(x_{n+1} – x_n) = 0
$$
But then it also follows that $y_n$ converges absolutely:
$$
\lim_{n\to\infty}|y_n| = \lim_{n\to\infty}|x_{n+1} – x_n| = 0
$$
Let's now fix some number $p \in \Bbb N$ and consider the following expression:
$$
\sigma_{n+p} – \sigma_n = \sum_{n+1}^{n+p} |y_k|
$$
Now consider the limit of RHS:
$$
\lim_{n\to\infty}\sum_{n+1}^{n+p}|y_k| = 0
$$
Then if follows:
$$
\lim_{n\to\infty}|\sigma_{n+p} – \sigma_n| = 0
$$
Thus $\sigma_n$ satisfies Cauchy's Criteria hence convergent, hence bounded.
I would like to ask for a verification of the proof above, and point to mistakes in case of any, or suggest a solution in case the above makes no sense. Thank you!
Best Answer
If $\{x_n\}$ is increasing, $$\sigma _n=x_{n+1}-x_1.$$ Since $\{x_n\}$ is bounded, then of course $\sigma _n$ is bounded...