Where is the mistake?  Statement: For $n\in \mathbb{N}_{0}$ is $2n=0$.  Bais: Show that the basis holds for $n=0$ : $2*0=0$.  Assumption: The statement is valid for all $k\leq n$ : $2*k=0$ for all $k \leq n$  Inductive Step: For $k = n+1$ is $k=a+b$ for two natural numbers $a,b \leq n$. It is $2(n+1) = 2a + 2b = 0+0=0$.  It is obviously wrong. I have a few ideas but I would like to be sure where the exact mistake is.  The first thing I noticed was that in this equoation $2(n+1) = 2a + 2b = 0+0=0$ the left side $2(n+1)$ is always positive but the right side isn't.  I cant find a mathematical mistake here so I thought the Assumption is wrong. I always thought the Assumption is valid for a specific n. It is is indeed freely selectable but specific. Is that right and is that the problem with this induction?   If it isn't - maybe it might be the fact that there aren't two natural numbers below 1 that $ a+ b =k$ and thet you cant use the assumption two times?
This is my last homework problem and I've been looking at it for a while. I cannot nail down what is wrong with this proof even though its obvious it is wrong based on its conclusion. Here it is:     Find the flaw in the following bogus proof by strong induction that   for all $n \in \Bbb N$, $7n = 0$.      Let $P(n)$ denote the statement that $7n = 0$.      Base case: Show $P(0)$ holds.      Since $7 \cdot 0 = 0$, $P(0)$ holds.      Inductive step: Assume $7·j = 0$ for all natural numbers $j$ where $0 \le j \le k$ (induction hypothesis). Show $P(k + 1)$: $7(k + 1) = 0$.      Write $k + 1 = i + j$, where $i$ and $j$ are natural numbers less than $k + 1$.  Then, using the induction hypothesis, we get $7(k + 1) = 7(i + j) = 7i + 7j = 0 + 0 = 0$. So $P(k + 1)$ holds.      Therefore by strong induction, $P(n)$ holds for all $n \in \Bbb N$.   So the base case is true and I would be surprised if that's where the issue is.  The inductive step is likely where the flaw is. I don't see anything wrong with the strong induction declaration and hypothesis though and the math adds up! I feel like its so obvious that I'm just jumping over it in my head.
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