Limit of Modulo Operation/Limit 2

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

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

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

Proof
As $y \to \infty$:

Therefore by the definition of modulo operation:

Hence the result.