Cauchy sequence: distance from point of convergence

Question

If $\{a_n\}$ is a Cauchy sequence and we know that $|a_i - a_{i+1}| < \varepsilon$ for some $i$. Is it possible to say anything about $|a_i - a|$, where $a$ is the point to which the sequence converges?

Answer 1

If the sequence is Cauchy, you know rather more than that: you know that for any $\epsilon>0$ there is an $n_\epsilon\in\Bbb N$ such that $|a_k-a_\ell|<\epsilon$ whenever $k,\ell\ge n_\epsilon$. It follows that $|a_k-a|\le\epsilon$ whenever $k\ge n_\epsilon$. To see this, suppose that there is a $k_0\ge n_\epsilon$ such that $|a_{k_0}-a|>\epsilon$, and let $\delta=|a_{k_0}-a|-\epsilon>0$. The sequence converges to $a$, so there is an $n_\delta\in\Bbb N$ such that $|a_k-a|<\delta$ whenever $k\ge n_\delta$. Now let $\ell=\max\{n_\epsilon,n_\delta\}$; then

$$\begin{align*} |a_{k_0}-a|&\le|a_{k_0}-a_\ell|+|a_\ell-a|\\ &<\epsilon+\delta\\ &=|a_{k_0}-a|\,, \end{align*}$$

which is absurd.