TITLE: Sequence that is neither increasing, nor decreasing, yet converges to 1
QUESTION [20 upvotes]: Give an example of a sequence which is neither increasing after a while, nor
decreasing after a while, yet which converges to 1. 
My solution: $1.01,\ .99,\ 1.001,\ .999,\ 1.0001,\ .9999,\ \text{etc}\dots$
Does that satisfy all the conditions? Also, judging by the instructions, do you think I would have to define that sequence? In which case, I could do $\{x_n\} = 1 + .01^n$ for odd $n$ and $1 - .01^n$ for even $n$ (which would change the sequence, but just increases the rate at which it approaches $1$). 
The definition of an increasing sequence used is the next term being bigger than OR equal to the preceding term. And dually for decreasing.

REPLY [2 votes]: Continued fraction representations of irrational numbers can be represented as an infinite sequence of rational partial convergents which oscillate above and below the number being represented.
Thus if we take $a_n$ to be the to be the nth partial convergent of any irrational number divided by that number, it will converge to 1 in an oscillatory manner.
$$
a_n = \frac{[\zeta]_n}{\zeta}, \zeta \in \mathbb{I}
$$
For $\zeta = \sqrt{2}$, we have
$$
a_0 = \frac{1}{\sqrt{2}} \approx 0.707\\
a_1 = \frac{3}{2\sqrt{2}} \approx 1.061\\
a_2 = \frac{7}{5\sqrt{2}} \approx 0.990\\
a_3 = \frac{17}{12\sqrt{2}} \approx 1.002\\
...
$$