I was looking over this problem and I'm not sure what's wrong with this proof by induction.
Here is the question:
Find the flaw in this induction proof.
Claim $3n=0$ for all $n\ge 0$.
Base for $n=0$, $3n=3(0)=0$
Assume Induction Hypothesis: $3k =0$ for all $0\le k\le n$
Write $n+1=a+b$ where $a>0$ and $b>0$ are natural numbers each less
than $n+1$Then $3(n+1) = 3(a+b) = 3a + 3b$
Apply Induction hypothesis to $3a$ and $3b$, showing that $3a=0$ and
$3b=0$. Therefore, $3(n+1)=0$
The statement they are trying to prove is clearly absurd but I'm having trouble with the logic in the proof by induction. It just seems like the person who wrote this proof used strong induction and applied the induction hypothesis to proof the implication.
Best Answer
The problem is it doesn't work for the first step after the base case: there do not exist $a \gt 0$, $b \gt 0$ such that $a + b = n + 1$ when $n = 0$. This is a variant of all horses are the same color.