Groups: Proof by Induction - Solved example

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Eli
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#1

Here are the problems. They will teach you how to prove mathematical statements by using induction method.

Problem 1

Let such that , for a fixed integer . Prove that positive, .

Proof

For , (true)

Let , we need to show that

So,

Suppose it is true for , i.e., , we show that it is also true for

Thus, , hence shown.


Problem 2

Let be a group and suppose that . Show that for any positive integer

Proof

The statement for is simply which is certainly true.

For



Hence the result is true for

Now, assume that the result holds for

Using this induction hypothesis, we have



The statement holds for so by induction it holds for all values of
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