What is the Infimum and Supremum of the Set $E= \{n/(n+1): n \in\mathbb N\}$?

Question

Let $$E = \left\{\frac{n}{n+1}: n \in\mathbb N\right\}.$$

I think that Infimum of this set exists and it is $1/2$ and the supremum of this set is $1$. But, I need a formal way to prove this that these are the infimum and supremum of this set along with that the fact that the values I have stated are actually the infimum and supremum of this set.

Answer 1

Hint: $\dfrac{n}{n+1}= 1- \dfrac{1}{n+1}.$

Answer 2

HINT

Note that

$$\frac{a_{n+1}}{a_n}=\frac{(n+1)^2}{n(n+2)}=\frac{n^2+2n+1}{n^2+2n}=1+\frac{1}{n^2+2n}>1$$

thus $a_n$ is strictly increasing.