My problem is to prove that cyclic groups of order 3 or greater must have at least 2 generators.

I think I found an answer but I'm worried that it's wrong.

Here's my proof.

In a group of order generated by there exists a "last element" . This element can be shown to generate every inverse element and therefore every element in the group.

For example

and so on until

Thus it is shown that in every cyclic group with the element generates the group as well as .

Is this valid?