If an inequality is true for all natural numbers, is it necessarily true for all real numbers inbetween?

Solution 1:

$x - \lfloor x \rfloor\le 0$, (where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$) is true for every $x \in \mathbb N$ but false for all other positive real numbers.

Solution 2:

Here is a more precise version of your question:

Question. If a statement of the form $\tau \geq \sigma$ (where $\tau$ and $\sigma$ are expressions built using only the operations $\{0,1,+,\times\}$) holds for $\mathbb{N},$ does it necessarily hold for $\mathbb{R}_{\geq 0}$?

Unfortunately, the answer is no, (thanks @Mathmo123).

Consider $x^2 \geq x$. This holds for all $n \in \mathbb{N}$ (in fact, for all integers), but not for the $r \in \mathbb{R}$ strictly between $0$ and $1$. Of course, a counterexample is rigorous proof of falsity.

Solution 3:

Cosine of $2n\pi$ is greater than zero for every integer $n$, but not for every real $n$...