Similar to the proof that all integers have interesting properties.
Suppose that there is a set of uninteresting integers. Consider the subset of uninteresting positive integers (if this is an empty set, then instead consider the set of all uninteresting negative integers and work with the negations of those). 1 is definitely an interesting number. It's the multiplicative identity, and has many other uses revolving around being unity. 2 is also an interesting number; it's the only even prime, for instance. Now, suppose all the numbers from 1 through n are interesting. n+1 must also be an interesting number! Why? Because if n+1 were uninteresting and all numbers from 1 to n were interesting, then n+1 would be the smallest uninteresting number, which is definitely an interesting property. Since this is a contradiction, then by strong induction, all natural numbers (and similarly, all integers) are interesting.