How do I show the greatest lower bound for this set is 17?

Question

Let $S = \{17 + \frac{1}{2n} : n \in \mathbb{N}\}$.

Prove that the greatest lower bound of $S$ is $17$.

What needs to be shown/proven?

Thanks in advance.

Answer 1

Suppose $x<17$ for some $x\in S$. Then for some $n$, $17+\frac{1}{2n}<17\implies \frac{1}{2n}<0$ which is a contradiction since $n$ is positive.

Now suppose that $17+\epsilon$ is a lower bound for $S$, where $\epsilon> 0$. We can pick $n$ such that $1/2n<\epsilon$ and thus $17+1/2n<17+\epsilon$. Thus $17+\epsilon$ cannot be a lower bound for $S$.