Limit of Modulo Operation/Limit 1

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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.

$\blacksquare$