Sequence that is neither increasing, nor decreasing, yet converges to 1

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.

Solution 1:

I think defining it the way you did is fine, but you could also use

$$s_n = 1+\left(-\frac{1}{10}\right)^n$$

which is a handy little trick to express alternating sequences nicely.

Solution 2:

Your solution works.

I like $1+\dfrac{\sin n}{n}$. But it's possible I'm weird.