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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?

This page last updated: May 26, 1998

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

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