Which conditions imply $\sup_n |\ln x_n| < \infty$?

Question

I want to find conditions which imply that $\sup_n|\ln x_n| < \infty$. Intuitively I think that $\inf_n x_n > 0$ and $\sup_n x_n < \infty$ should be enough, but I don't know how to write it formally. Are there any properties I can use?

Answer 1

You know that $|\log x_n| \le A$ for all $n$ if and only if $$ -A \le \log x_n \le A \quad \Longleftrightarrow \quad e^{-A} \le x_n \le e^A. $$

Therefore, you see that

• If $A = \sup_{n \in \mathbb N} |\log x_n| < \infty$, then $e^{-A} \le \inf_{n \in \mathbb N} x_n$ and $\sup_{n \in \mathbb N} x_n \le e^A$.
• If $\inf_{n \in \mathbb N} x_n > 0$ and $\sup_{n \in \mathbb N} x_n < \infty$, let $B = \max \{ -\log(\inf_{n \in \mathbb N} x_n), \log(\sup_{n \in \mathbb N} x_n) \}$. Then $\sup_{n \in \mathbb N} |\log x_n| \le B$.

Hope that helps,

Answer 2

Well, if you start with those assumptions, then assume the $sup_n |\ln x_n|=\infty$ and you can quickly get a contradiction. That says that there exists a subsequence that is unbounded. That means you must have an infinite amount of $\ln x_n$ that are unbounded to $\infty$, or that arbitrarilly approach 0. If That would contradict your two assumptions.

That shows your two conditions are sufficient. To show they are necessary, assume one of them is not true, and you immediately can get a subsequence of $\sup_n |\ln x_n|$ that is unbounded. Hence they are necessary and sufficient