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.

Yes, your sequence satisfies all required conditions.
Another example sequence:
$$a_n = \begin{cases}
1+\frac 1n & \text{if } \log_{10}n \in \mathbb N \cup \{0\} \\
1 & \text{otherwise}
\end{cases}$$
has terms:
$$\begin{align}
a_1 & = 2 \\
a_{10} & = 1.1 \\
a_{100} & = 1.01 \\
a_{1000} & = 1.001 \\
a_{10000} & = 1.0001 \\
\ldots
\end{align}$$
and all remaining terms equal $1$.
