What is a sequence that gets closer and closer to a without converging to a?

Question

Find a sequence ${a_n}$ and a real number $a$ so that $|a_{n+1}-a| < |a_n-a|$ for each n, but $a_n$ does not converge to $a$. So the sequence gets closer and closer to $a$ without converging to $a$.

I have seen many questions posted here asking how to find a sequence that converges to $a$ without getting closer and closer to $a$, so this question is kind of the opposite. My intuition is telling me that I need to find a function that increases slower and slower but continues to increase. Any and all help is appreciated- thank you!

Answer 1

For example, $$a_n=\frac1n\quad\hbox{and}\quad a=-2018\ .$$

Answer 2

You can simply take $a=0$ and $a_n=1+\frac1n$.

Answer 3

Hint: If a sequence converges to $1$ from above, the distances between each of its terms and $0$ decrease.

Answer 4

Your intuition appears to be leading you in a good direction. An increasing sequence is a good thing to try.

Note that an increasing sequence that converges to a number has a property you're looking for: it increases slower and slower.

Try to construct an actual increasing, converging sequence $a_n.$ Then consider some of the numbers that $a_n$ does not converge to. Could any of them be the number $a$ you need?