Question

If x > 0, show that there exists n ∈ N such that 1/n < x.

Answer 1

Here we have $(x\in\R)>0$ . Which implies $\frac{1}{x}>0$ .

Now, using Archimedean property, we have $n\in \N$ such that,

$$0<\frac{1}{x}<n\\~\\ \Rightarrow n>\frac{1}{x}\\~\\ \Rightarrow x>\frac{1}{n}$$

Hence, $\exists n\in \N$ such that, $x>\frac{1}{n}$