Book error: $\forall x:\mathbb R, x\gt 2 \Rightarrow x \ge 3$.

Question

I have a problem understanding the logic behind the use of an implication in the following question that comes from a text book that shall remain nameless...

Is the following true or false: $$\forall x \in \mathbb R, x\gt 2 \Rightarrow x \ge 3$$ The correction from the book is: True. If a real number is superieur to 2, it is superieur or equal to 3

I've contacted the author to query this, and he replied that it's an implication and not an equivalence, so it's true in $\mathbb R$.

I believe that it's false, because if $x = 2.5$, then $x\gt 2$ and $x \ge 3$ is false and the implication fails.

lulu has asked what book this question comes from. Given that everyone so far agrees that the book is wrong, here is the link to it on amazon in France (sorry it's a french book): http://amzn.eu/eVxvUR1 The question appears on page 20 of the Look inside preview (Feuilleter in French), the correction on page 37.

Thanks for the replies everyone - the author's reply had me wondering "what was I missing".

Answer 1

You are right. $x=2.5$ in fact makes $x>2$ true but $x \ge 3$ false and hence it is a valid counter example.