Finding least upper bound and greatest lower bound of the set

Question

I am given the following set:

$A = \left\{ \frac{1}{n} + (-1)^{n}: n \in \mathbb{N} \right\}$

I am asked to find the greatest lower bound and least upper bound, but am having difficulty. I tried to split the set up into when $n$ is even and when $n$ is odd. So, when $n$ is even, $A = {(1/n) + 1}$ and when $n$ is odd, $A = (1/n) - 1$.

Where does one go from here?

Answer 1

Each element of $A$ is smaller than or equal to $\frac32\left(=\frac12+(-1)^2\right)$, which is an element of $A$. So, $\sup A=\frac32$. On the other hand, each element of $A$ is greater than $-1$. But, if $-1<a<0$, there is some odd number $n$ such that $\frac1n<a+1$, and then $(-1)^n+\frac1n<a$. Therefore, $a$ is not a lower bound of $A$ and so $-1$ is the greatest lower bound. So, $\inf A=-1$.

Answer 2

Hint: It seems clear: if you denote $a_n=\frac1n+(-1)^n$, all terms with even index are positive, $>1$ and make up a decreasing sequence, whereas all terms with odd index are nonpositive and $>-1$ and also make up a decreasing sequence. Can you conclude from these observations?