Limit of Modulo Operation/Limit 2

Theorem

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

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

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

As $y \to \infty$:

$\ds 0$

$\le$

$\, \ds x \, $

$\, \ds < \, $

$\ds y$

$\ds \leadsto \ \ $

$\ds 0$

$\le$

$\, \ds \frac x y \, $

$\, \ds < \, $

$\ds 1$

$\ds \leadsto \ \ $

$\ds $

$\, \ds \floor {\frac x y} \, $

$\, \ds = \, $

$\ds 0$

Therefore by the definition of modulo operation:

$\ds \lim_{y \mathop \to \infty} x \bmod y$

$=$

$\ds \lim_{y \mathop \to \infty} x - y \floor {\dfrac x y}$

$\ds $

$=$

$\ds \lim_{y \mathop \to \infty} x - y \cdot 0$

$\ds $

$=$

$\ds x$

Hence the result.

$\blacksquare$