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Remember, in order to prove a statement of the form "if A, then B", what you do is assume A and derive B from it.

In our case, A is the statement *S*(*k*): "in any group of *k* people,
everyone has the same age", and B is the statement
*S*(*k*+1): "in any group of *k*+1 people,
everyone has the same age."

Step 4 says that all we have to do to complete the proof is assume A is true and try to derive B from it. Therefore, it is legitimate to use A and treat it as true in the process of proving B.

That's exactly what we're doing here. At this stage in the proof, we have two things are disposal:

- we are assuming that, in every group of
*k*people, everyone has the same age, and - we now have a particular group
*G*of*k*+1 people, and are trying to use the above assumption to prove that everyone in*G*has the same age.

Within *G*, there happens to be a group of *k* people (namely,
the group of
everyone in *G* except for the one person *Q*).

Since we are operating under the assumption that,
in every group of *k* people, everyone has the same age,
it follows that everyone in this smaller group within *G* has the same
age.

And, it is perfectly legitimate to use this knowledge about this smaller
group within *G*
to try to show something about *G* itself,
which is what the subsequent steps in the proof attempt to do.

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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