# Thread: Example of sequence with given property

1. ## Example of sequence with given property

What examples would be true for the following?

1. The sequence is convergent but not monotone
2. The sequence is strictly decreasing but not convergent
3. The sequence is neither bounded nor monotone
4. The sequence is bounded below, not bounded above, and contains an infinite number of negative terms

2. Come on, these can't be that hard.

Originally Posted by summerset353
What examples would be true for the following?

1. The sequence is convergent but not monotone
$\displaystyle a_n=\frac{(-1)^n}{n}$

2. The sequence is strictly decreasing but not convergent
Think about, $\displaystyle a_n=-\left(1+\frac{1}{n}\right)^n$ for example in $\displaystyle \mathbb{Q}$

3. The sequence is neither bounded nor monotone
$\displaystyle n(-1)^n$

4. The sequence is bounded below, not bounded above, and contains an infinite number of negative terms
how about $\displaystyle a_n=\begin{cases} n &\mbox{if}\quad n\text{ is even}\\ \frac{-1}{n} & \mbox{if} \quad n\text{ is odd}\end{cases}$. Clearly $\displaystyle a_n\geqslant1$ but is not bounded above and contains infinitely many negative terms.

3. Even easier for 2: $\displaystyle x_n=-n$.

4. Originally Posted by redsoxfan325
Even easier for 2: $\displaystyle x_n=-n$.
I know. I just was trying to make realize that saying a sequence is convergent without speaking of the ambient space is just...well wrong.

5. Lol...I agree.