How to prove that $a_n$ is Cauchy if $[a_n, b_n]$ is a sequence of nested intervals such that $(b_n − a_n) → 0$ as $n → ∞$.

Question

ATTEMPT

Suppose $[a_n, b_n]$ is a nested sequence of closed intervals whose lengths tend to zero as $n$ → ∞.

The Completeness Axiom guarantees that any nonempty set of real numbers $S$ that is bounded above has a least upper limit.

To show that $a_n$ is Cauchy, $b_n$ in the expression $|b_n − a_n|$ can be replaced by $a_m$ for $m$ ≥ $n$ to have $|a_m − a_n|$, which is even smaller than |$b_n$ − $a_n$|.

I am confused about how to proceed further.

Any help will be highly appreciated. Thank you.

Answer 1

I think you have the right idea, here's how I would extract the conclusion from it:

Let $\varepsilon > 0,$ then since $|b_n - a_n| \to 0$ as $n \to \infty$, there exists some $N \in \mathbb N$ such that for all $n$ larger than $N$, $$ |b_n - a_n| < \varepsilon. $$ Now for all $m$ larger than $n$, using your observation we have $$ |a_m - a_n| \leq | b_n - a_n | < \varepsilon, $$ since $a_n \leq a_m \leq b_n$. This is precisely what we need to conclude $(a_n)_{n \in \mathbb N}$ is Cauchy, so we are done.