Let $M$ be the set $M = ( \frac{1 + 2n^2}{1 + n^2} \in \mathbb{R} : n \in \mathbb{N_0} ) \subset \mathbb{R}$. Find Inf, Sup, Min, Max

Question

Let $M$ be the set

$$M = ( \frac{1 + 2n^2}{1 + n^2} \in \mathbb{R} : n \in \mathbb{N_0} ) \subset \mathbb{R}$$

Determine Infimum, Supremum, Minimum and Maximum of $M$, if those exist

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Minimum: $n = 0$ gives $1 / 1 = 1$

Maximum: $$\lim_{n \to \infty} \frac{1/n^2 + 2}{1/ n^2 + 1} = 2/1 = 2$$

But it never really reaches this value, because it tends to it. So there is no maximum.

The Infimum is $1$, and the Supremum is $2$

Is that correct ? We have no solutions to this exercise, please give me your feedback. Thank you !

Answer 1

Note $\frac{2n^2 + 1}{n^2+1} = \frac{2n^2 + 2 -1}{n^2+1} = 2 - \frac{1}{n^2+1}$.
From this, it is clear that minimum (hence infimum) is at $n=0$ ie. $ \min M = \inf M = 1$
$2$ is an upper bound for $M$ and difference between $2$ and elements of $M$ is of the form $\frac{1}{n^2+1}$, which can be made sufficiently small but never zero. So, $2$ indeed the supremum of $M$ but at the same time it can't be the maximum.