A counterexample of induction on integers?

Question

What could be an example of a property $P(n)$ pertaining to an integer $n$ such that $P(0)$ is true, and that $P(n)$ implies $P(n++)$ for all integers $n$, but that $P(n)$ is not true for all integers $n$?

(Reference: http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf, Page 33, The paragraph after lemma 18).

Answer 1

Any property that satisfy exactly the integers contained in some interval $[a,\infty)$, where $a\le 0$. For example:

• To have a real square root.
• To satisfy the inequality $x^3+100>0$.
• To be non-negative.

Etc.