# Number less than Integer iff Floor less than Integer

## Theorem

Let $x \in \R$ be a real number.

Let $\floor x$ denote the floor of $x$.

Let $n \in \Z$ be an integer.

Then:

$\floor x < n \iff x < n$

## Proof

### Necessary Condition

Let $x < n$.

By definition of the floor of $x$:

$\floor x \le x$

Hence:

$\floor x < n$

$\Box$

### Sufficient Condition

Let $\floor x < n$.

We have that:

$\forall m, n \in \Z: m < n \iff m + 1 \le n$

and so:

$(1): \quad \floor x + 1 \le n$

Then:

 $\ds x$ $<$ $\ds \floor x + 1$ Definition of Floor Function $\ds$ $\le$ $\ds n$ from $(1)$

$\Box$

Hence the result:

$\floor x < n \iff x < n$

$\blacksquare$