# Prove that $f(x) = \cos(1/x)$ is not periodic without using calculus

algebra-precalculusproof-verificationproof-writingtrigonometry

Given a function $$f(x)=\cos\frac{1}{x}$$ Prove that it is not periodic.

I've started with assuming that $f(x)$ is periodic and hence $f(x) = f(x+2\pi)$ since $\cos$ hat period $T = 2\pi$. This gives:

$$\cos\frac{1}{x} = \cos\frac{1}{x+2\pi}$$

Given the above we may take some value x and check whether the equation holds for that value. Let's say $x=\frac{\pi}{2}$, then
$$\cos\frac{2}{\pi} = \cos\frac{2}{5\pi}$$
which is not true. Hence $f(x) = \cos\frac{1}{x}$ is not periodic. I've found similar questions here and here but they use derivatives for the proof.

Can I somehow prove the above in a general way without taking any exact value for $x$ and without using calculus?

Assume $$\cos\frac1x=\cos\frac1{x+T}.$$
$$\frac1x=\pm\frac1{x+T}+2k\pi,$$
$$k=\frac{\dfrac1x\mp\dfrac1{x+T}}{2\pi}.$$
But the RHS cannot be an integer for all $x$, a contradiction.