Suprema and infima of subsets of Real Numbers and The Completeness Property of R.

Question

Let S = {1$\ $-$\ $$\frac{(-1)^n}{n}$:n ∈ N}. Find the Infimum and Supremum of S.

When I simply put the values of n, I find that the greatest element is 2 and the least element is $\frac{1}{2}$. How can this be solved analytically?

Answer 1

The same argument can be used to show both assertions. Let $f(n) = 1-\frac{(-1)^n}n$. Notice that $\big\lvert\frac{(-1)^n}{n}\big\rvert = \big\lvert\frac1n\big\rvert < 1$ if $n\geq 2$. Therefore, for every $n$ odd, $f(n)\leq f(1)$ and for every even $n$, we have $f(n)\geq f(2)$.

Therefore, $\inf S = f(2)$ and $\sup S = f(1)$.

Answer 2

2 séquencies: - Even sequence increasing, begining with the value 1/2 ( for n=2 ) and finishing with the value 1 ( at infinity ) - Odd sequence, which is decreasing, begining with the value 2 ( for n= 1) and finishing with the value 1 ( at infinity ) So we have the gretest element 2 and thés mal l'est element 1/2.