A lot of the time in lectures, my professors prove (by induction) an inequality (e.g. $(1+x)^n \geq 1+nx$) in the natural numbers (or any subsets thereof), and I've noticed (not rigourously; only by graphing the functions) that such statements are also true for all real numbers inbetween.
Another example is that exponential growth beats polynomial growth.
My question is:
If an inequality is true for all $n \in \mathbb{N},$ does it necessarily follow that the same inequality is true for all $n \in \mathbb{R^+}$?
I'm not in the market for a rigourous proof; (if the answer's no) just a counter-example, or (if the answer's yes) an intuitive reason why this is the case.