This should be an easy question. Yet, the provided solution confuses me.
The question comes from "Understanding analysis" by S. Abbot, 2nd edition (Exercise 1.2.11).
Negate the statement. Make an intuitive guess as to whether the claim or its negation is the true statement.
(b) There exists a real number $x > 0$ such that $x < 1/n\;\;\forall n \in \mathbb{N}$.
The provided solution says:
The solution seems correct, apart from: shouldn't the negation be with $\exists n \in \mathbb{N}$, i.e.:
$$\forall x >0 \;\; \exists n \in \mathbb{N}: x \geq 1/n$$
?
Best Answer
This is why putting quantifiers at the end of a formula is a bad practice. It creates ambiguity. The statements
are not equivalent. The second one is obviously false, however it's more likely to interpret your formulation as the second statement. Undoubtedly, the first statement is what's actually meant. For proving by contradiction, we need its negation which goes as follows: $$(\exists n \in \mathbb{N})(\forall x > 0)\left(x \ge \frac{1}{n}\right)$$