Confused about my professor's answer for an inf and sup question

Question

For $x_n=\frac{(-1)^n}{n}$, my professor said that the inf for when n is even is $\frac{-1}{n+1}$ and the sup is $\frac{1}{n}$. How is the inf a negative number since $(-1)^n$ where n is an even number is always positive?

Answer 1

What your professor said, if that is exactly what he said, makes zero sense. The infimum and supremum are fixed number and should not include $n$. If you take $n$ to be even, as you say, you always have positive terms. The supremum is $\frac{1}{2}$ which is for $n=2$. You can see this, since for all larger $n$, the sequence becomes smaller. The infimum would be $0$. Now, for the odd case, you always have negative numbers, so the infimum would be $-1$ and the supremum $0$.

The whole idea behind the supremum and the infimum of a sequence, is that you consider all the terms of the sequence at once, and want to find the largest and smallest one. However, sometimes, there is no largest or smallest element, so the supremum or infimum do not actually belong in the set. Here, you can see this since $0$ is not an actual element of the sequence.

Answer 2

Your teacher is flatly wrong ! the $\inf(x_n) = 0$ when $n$ is even, and $\sup(x_n) = \dfrac{1}{2}$ when $n$ is also even