Prove that function $f(n) = \cos(1/n)$ is monotonically decreasing? Ho can we prove that this function

$$f(n) = \cos(1/n)$$

is monotonically decreasing? I computed derivative of this function and got:

$$f'(n) = \frac{\sin(1/n)}{n^2}$$

and it is positive, but should be negative, because function $f(n) = \cos(1/n)$ is monotonically decreasing.
Is $n$ supposed to be a positive integer? If so you should probably call $f$ a sequence instead of a function, to prevent confusion.

Assuming that yes you are talking about the sequence $\cos(1/n)$: That sequence is not decreasing, it's increasing.

In detail: The sequence $1/n$ is decreasing. And $0&lt;1/n&lt;\pi$, which means that $\cos$ is also decreasing on the relevant range. A decreasing function of a decreasing sequence is increasing.
Remember that the derivative of $1/n^2$ is negative.