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This step is not the source of the fallacy.This step is simply stating what happens in an induction argument.
The principle of induction says that, if the following two things
then S(n) is true for all n. (For more details, see the
brief summary of induction).
- S(1) is true, and
- For all natural numbers k: if S(k) is true, so is S(k+1),
This step in the proof is simply asserting that part 1 above
has already been proven (this follows from step 2), and that therefore
proving part 2 is enough to prove that S(n) is true for all n.
Why don't you go back to the list of steps in the
proof and see if you can identify which one
is wrong, now that you know it isn't this one?
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