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A set $A = \left\{(-1)^n + \frac{1}{m} : n,m \in \Bbb N\right\} \cup \{-1\}$ is given.

a) I shall find out and justify what the supremum an infimum is.

b) Is the sup a maximum or the inf a minimum.

a) For the sup, I find out $n=2$ and $m=1$ because then I have $1+1=2$. For the inf, I have $\displaystyle \lim _{m\to\infty} (1/m) = 0$ and $n = \text{odd number} \implies 0-1=-1$.

Now my questions are: Is that correct? How can I now find out if there is a max or min? Can I say for the sup that $2$ and $1$ are elements of $\Bbb N$ so the sup is a max and for the inf same? And what about the $\{-1\}$ in the set, what does this $\{-1\}$ mean for the inf, sup, min and max?

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    For a very similar problem(and solution) see [here](http://math.stackexchange.com/questions/810064/supremum-of-two-subtracted-fractions-less-than-one/810095#810095).2014-06-04

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