logic - Proof by deduction - implications

Currently trying to explain some maths to a friend.

He has taken a statement $x^2 + 4 > 2x$ and tried to prove this is true for all $x$.

His proof is $x^2+4>2x \Rightarrow x^2-2x + 4 > 0 \Rightarrow (x-1)^2 + 3 > 0$ which is true so the original statement is true.

However this starts at the wrong place and the implication goes in the wrong direction. So I think it’s wrong and I can’t seem to convince him of this or find some basic examples to illustrate the point that statement X $\rightarrow$ true statement doesn’t mean that X is true....

So can anyone explain to me why it’s wrong using some basic counterexamples perhaps so I can have the knowledge to explain why it is wrong...

Thanks

Answer

Your friend is correct, the subtlety is that all his steps are reversible, so a clear way to put it is as:

$$x²+4>2 \iff x²-2x+4>0 \iff (x-1)²+3>0$$
This way the truthiness of the last statement implies the same for the first.
But you are correct to be cautious, a case where things would go wrong is with squares. For example:
$$x=1 \Rightarrow x² = 1 \Rightarrow x=1~\text{or}~x =-1$$
The last sentence is true if $x=-1$, but the first would be false.