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Home PageHowever, 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).