checking the answer for infimum and supremum of a set

Question

I want to find the infimum and supremum of the set $$S=\left\{\frac{3n+2}{2n+1}\mid n \in \mathbb N \right\}$$

I found $\inf S=\frac32$ and $\sup S=\frac53$. Is that correct?

Answer 1

Note that $$\frac{3n+2}{2n+1}=\frac{3n+\frac32+\frac12}{2n+1}=\frac{\frac32(2n+1)+\frac12}{2n+1}=\frac32+\frac{\frac12}{2n+1}.$$ Further note that the function $\Bbb N\to\Bbb R$ given by $$n\mapsto\frac32+\frac{\frac12}{2n+1}$$ is strictly decreasing, and its range is exactly the set in question. Show that the set has a maximum (which is the supremum), and find the limit of the function as $n\to\infty$ to get the infimum. You may need to prove that this limit is the infimum.