So I have recently come to understand the mechanics behind statements like "Prove that $a_n > b$ for all but finitely many $n$," but I am struggling with this question.
Find a sequence $\{a_n\}$ and $a \in \mathbb R$ so that $a_n \to a$ but so that the inequality $|a_{n+1} - a| < |a_n-a|$ is violated for infinitely many $n$.
The moral we are supposed to draw from a question like this is that the sequence can converge without getting "closer and closer" but that is not what I draw from this statement.
I see this question as talking about a sequence like one from this pdf which looks like this
$8, 1, 4, 1/2, 2, 1/4, 1, 1/8, 1/2, 1/16, \ldots \tag{*}$
This sequence "bounces around" like I would imagine the one we are supposed to come up with for the question, but I don't know what the explicit formula that generates the one in $(*)$.
So I'm a little stuck. Any starting hints would really help.