Write a negation of the following statement without using words of negation: A bounded real function cannot be surjective." Which is true, the statement, or its negation? Justify your answer.
ATTEMPT: An unbounded real function can be surjective. And for the second part I think the negation is True since if a function is bounded, how can it ever reach all the elements in the real numbers due to its restrictions.
Am I right?
Let $f:\mathbb{R}\to \mathbb{R}$. Your statement can be write as follow:
$$\exists M>0\ \forall x\in\mathbb{R}(|f(x)|<M\Rightarrow \exists a\in\mathbb{R} \ \forall y\in\mathbb{R},\ f(y)\neq a)$$ then the negation (accordingly to the logic rules) is:
$$\exists M>0\ \forall x\in\mathbb{R}(|f(x)|<M\ \wedge\ \forall a\in\mathbb{R} \ \exists y\in\mathbb{R},\ f(y)= a).$$