Consider the statement:
Any set of coins chosen from a box will have the same denomination.
This is obviously not a true statement, since in general a set of coins will contain more than one denomination. Yet I have an argument that seems to prove the statement is true by mathematical induction. What is the flaw in the "Proof"?
"Proof": Let P(n) be the statement "Any set of n coins chosen from a box all have the same denomination."
P(1) is clearly true.
Suppose that P(k) is true. Consider any set of k+1 coins: Remove one coin. Since P(k) is true, the remaining coins all have the same denomination: Now put that coin back, and remove a different coin. Again, since P(k) is true, the remaining k coins all have the same denomination. Therefore, all k+1 coins have the same denomination, that is, P(k+1) is true. Thus, by the principle of mathematical induction, P(n) is true for all n.