Supremum and Infimum of $\lbrace \frac{n+1}{n}|n\in \mathbb{N}\rbrace$

Question

For $S=\lbrace \frac{n+1}{n}|n\in \mathbb{N}\rbrace$, determine whether $S$ is bounded above, bounded below, bounded, or neither.

If $S$ is bounded above, determine $sup(S)$, and show whether or not $sup(S)\in S$. If S is bounded below, determine $inf(S)$, and show whether or not $inf(S)\in S$.

Answer 1

We know $0 \lt1/n \le 1$. So we have $1 \lt1/n + 1 \le 2$ . From here we can conclude $\sup(S) = 2$ and $\inf(S) = 1$ . So $S$ is bounded (It means $S$ is bounded above and below) .

Answer 2

Since $1/n \le 1$ with equality at $n=1$, we know $\sup\{S\} = 2$ which is achieved at $n=1$.

As $1/n \to 0$ as $n \to \infty$ and $1+1/n > 1$, it is reasonable to guess $\inf\{S\} = 1$. Let us check this is the case. By definition, if $\inf\{S\}=1$, for every $\varepsilon > 0$, there exists some $s \in S$, such that $s \le 1+\varepsilon$. This is clearly true by taking $n \ge 1/\varepsilon$. So $\inf\{S\} = 1$.

Since $\inf \{S\} $ amd $\sup\{S\}$ are both finite, $S$ is bounded above and below.