Limit of $f(x) = x \bmod k$

Question

I'm trying to prove that a function $f(x)$ tends to infinity when $x$ rises.

Clearly, I used limit to do so.

The problem is, $f(x) = x\ mod\ k$, in which mod is the division's residue of $x$ by $k$.

$$\lim_{x\to\infty} x\ mod\ k\ = \infty$$

Note that $k$ is an arbitrary constant.

I don't know how to express $mod$ in a mathematical way so to prove this.

Can you help me?

Answer 1

Your thesis is wrong: $f(x+k)=f(x)$ by definition. Thus, since the function is periodic and not constant, the limit you are looking for does not exists

Answer 2

Work with small examples when one is stuck.

For example, if you let $k$ to be $2$, the relevant sequence would be $f(0)=0,f(1)=1,f(2)=0,f(3)=1,f(4)=0,f(5)=1,\ldots$ of which the limit does not exists.