Limit of Modulo Operation/Limit 1

Theorem
Let $x$ and $y$ be real numbers.

Let $x \bmod y$ denote the modulo operation.

Then $\ds \lim_{y \mathop \to 0} x \bmod y = 0$.

Proof
By Range of Modulo Operation for Positive Modulus and Range of Modulo Operation for Negative Modulus we have:
 * $-\size y < x \bmod y < \size y$

The result follows from the Squeeze Theorem for Functions.