I am trying to express contrapositive of the statement "For all positive real number $x$, there exists an integer n such that $\frac{1}{n} < x.$" in formal language.
The universe for both $x$ an $n$ is the set of all real numbers.
I am confused whether the expression for the contrapositive of the statement in formal language should be
\begin{align}
\forall x,(\forall n,(n\in \mathbb{Z} \rightarrow \frac{1}{n} \ge x) \rightarrow (x \notin \mathbb{R}^+)).
\end{align}
or
\begin{align}
\forall x,(\forall n,(n\in \mathbb{Z} \wedge \frac{1}{n} \ge x) \rightarrow (x \notin \mathbb{R}^+)).
\end{align}
Best Answer
The statement, expressed as an implication with free variable $x$, is $$ x\in\mathbb{R}^{+}\rightarrow \exists{n}:\left(n\in\mathbb{Z}\;\wedge\;\frac{1}{n}<x \right). $$ Its contrapositive, then, is $$ \neg\exists{n}:\left(n\in\mathbb{Z}\;\wedge\;\frac{1}{n}<x \right)\rightarrow x\not\in\mathbb{R}^{+}, $$ or $$ \forall{n}:\left(n\in\mathbb{Z}\rightarrow\frac{1}{n}\ge x \right)\rightarrow x\not\in\mathbb{R}^{+}. $$