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This step is not the source of the fallacy.This step is correctly stating what it means to prove
an "if A, then B" type of statement: you assume A, and deduce
B from it.
However, there is something missing from this step. It hasn't said
what k is. Remember, as part of proving by induction that something
is true for all natural numbers, one must show for each
natural number k
that the truth of S(k) implies the truth of S(k+1).
Obviously, one can't do this separately for each k, as there are
infinitely many of them! But one can construct an argument that works
no matter what k is, and that's enough to establish it for all k.
So, strictly speaking, this should be stated in the step, and it should
read something like "We can do this by letting k be an arbitrary
natural number, then (1) assuming ...". The phrase "letting
k be an arbitrary natural number" means that k is unspecified and
the following argument is supposed to be valid no matter what k is
(as long as it's a natural number, of course).
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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