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 $0<1/n<\pi$, which means that $\cos$ is also decreasing _on_ the relevant range. A decreasing function of a decreasing sequence is increasing.