There is a common mistake people make when making proofs you did the same mistake.

"You assume something is true, you reach the conclusion that is definitely true. Then you cannot conclude that you hypothesis was true".

1)You assume

2)You arrive at

Now statement (1) was your hypothesis, you assumed it was true. And statement (2) was definitely true. You cannot conclude that (1) is true, it is inconclusive.

It was demonstated with my faulty proof

.

1)I assumed that

2)I arrived at

Now statement (2) was definitely true. I cannot conclude that statement (1) is true. Namely, that

it clearly does not work here.

This is called taking the converse of a statement.

Everything you did was good and well-done but it had a problem based on mathematical logic. My proof that is exatly like you only backwards is not based on the assumption of the true validity of the hypothesis statement.