Find supremum and infimum of the sequence $$a_n = \frac{2n^2}{4n^2+1},\:\:n\:\in \mathbb{N}.$$
I thought the supremum is $1/2$ and there is no infimum because the sequence gets close to ${0}$, but $0\notin \mathbb{N}$.
In addition, I need to find $n$ that $a_n\in \left(S-0.0080,S\right)$ and $a_n\in \left(I,I+0.0080\right)$.
Well, since the $a_n$ are interpreted as elements of $\mathbb R$ and not of $\mathbb N$ (since they are not even in $\mathbb N$) there will be an infimum. The sequence does not get close to $0$, in fact $$\lim_{n \rightarrow \infty} a_n = \frac{1}{2}.$$ Since the sequence is increasing, the supremum is $1/2$ you will find the infimum by computing the first term.