The set you're asking about is:

$S = \left\{ 3 + \frac{1}{n} \mid n \in \mathbb{N} \right\}$

Where $\mathbb{N}$ is the set of natural numbers. To find the infimum (inf) and supremum (sup) of this set, we need to analyze the behavior of the terms as $n$ changes.

Analysis of the set:

• The expression for each term is $3 + \frac{1}{n}$.
• As $n \to \infty$, $\frac{1}{n} \to 0$, so the terms approach 3 from above. Hence, the set gets arbitrarily close to 3 but never reaches it for finite $n$.
• When $n = 1$, the term is $3 + \frac{1}{1} = 4$.
• For larger values of $n$, the terms decrease, but they will always be greater than 3.

Infimum (inf):

• The infimum is the greatest lower bound of the set. Since the terms approach 3 as $n \to \infty$, but never actually reach 3, the infimum of the set is 3. This is because for all $n$, $3 + \frac{1}{n} > 3$, and 3 is the smallest possible value that the terms get arbitrarily close to.

Thus, the infimum is:

$\inf S = 3$

Supremum (sup):

• The supremum is the least upper bound of the set. The largest value occurs when $n = 1$, which gives $3 + 1 = 4$.
• Since all other terms are smaller than 4, the supremum of the set is 4.

Thus, the supremum is:

$\sup S = 4$

Conclusion:

• The infimum of the set is $3$.
• The supremum of the set is $4$.

Let me know if you'd like a more detailed explanation or if you have any questions!

Here are 5 related questions you might find interesting:

1. How do limits play a role in determining the infimum and supremum of sequences?
2. Can the supremum or infimum of a set be an element of the set?
3. What is the difference between the infimum and minimum, or supremum and maximum of a set?
4. How does the concept of a convergent sequence relate to infimum and supremum?
5. What happens if we consider the set for non-natural numbers (like rational or real numbers)?

Tip: Always check if the sequence has a limit as $n$ increases. The infimum of a convergent sequence can often be the limit of the sequence if it's the smallest value the terms approach.