Simple Examples where Base Case of Induction is non-trivial? I want to explain the importance of the base case of induction to a 10-year-old kid. But I am finding it difficult to find examples where solving the base case is non-trivial.
For example, the sum of $n$ natural numbers from $1$ to $n$, is $T(n) = \frac{n(n+1)}{2}$. Its base case would be $T(1) = 1$, which is trivial to see. I do not want examples like this. Neither I want complicated examples. Can somebody suggest some easy but non-trivial examples here?

Note: The same question has been asked before here. But the examples therein are difficult to understand except this one. But I think the base case of this example is not provable. I am looking for non-trivial base case examples that are solvable. I hope you understand my point.

How about something completely made-up and informal?
Theorem: At every age $n$, a person owns infinitely many ice cream cones.
Proof: Proof by induction: Assume every person of age $n$ owns infinitely many ice cream cones. A person of age $n+1$ must have had infinitely many ice cream cones one year prior, and assuming a lower bound to the volume of an ice cream cone and an upper bound to the speed of ice cream consumption, you can only eat finitely many ice cream cones in a year. Therefore the supply could only have shrunk by a finite amount, and remains infinite at age $n+1$. QED.
Obviously, the flaw of this proof is the fact that people are not born with an infinite amount of ice cream, i.e. the base case is not fulfilled.
