To find the supremum of the set $S = \{ 2 - \frac{1}{n^2} : n \in \mathbb{N} \}$, we start by analyzing the terms in the set.

Step 1: Identify the elements of S

For natural numbers $n$:

• When $n = 1$, $2 - \frac{1}{1^2} = 2 - 1 = 1$
• When $n = 2$, $2 - \frac{1}{2^2} = 2 - \frac{1}{4} = \frac{7}{4} = 1.75$
• When $n = 3$, $2 - \frac{1}{3^2} = 2 - \frac{1}{9} = \frac{17}{9} \approx 1.89$
• When $n = 4$, $2 - \frac{1}{4^2} = 2 - \frac{1}{16} = \frac{31}{16} = 1.9375$

As $n$ increases, $\frac{1}{n^2}$ decreases, causing $2 - \frac{1}{n^2}$ to increase.

Step 2: Determine the limit as $n$ approaches infinity

Taking the limit as $n \to \infty$:

$\lim_{n \to \infty} \left( 2 - \frac{1}{n^2} \right) = 2 - 0 = 2$

This indicates that the values in $S$ can get arbitrarily close to 2, but never actually reach it, since $\frac{1}{n^2}$ is always positive for $n \in \mathbb{N}$.

Step 3: Establish the supremum

1. Upper Bound: We claim that 2 is an upper bound for $S$. For all $n \in \mathbb{N}$:

$2 - \frac{1}{n^2} < 2$

This shows that no element in $S$ can equal or exceed 2.

2. Least Upper Bound: To show that 2 is the least upper bound (supremum), we can argue that for any $\epsilon > 0$, there exists an $n$ such that:

$2 - \frac{1}{n^2} > 2 - \epsilon$

This means we can find an $n$ sufficiently large such that $\frac{1}{n^2} < \epsilon$. Specifically, if we choose $n$ such that $n > \frac{1}{\sqrt{\epsilon}}$, we have:

$\frac{1}{n^2} < \epsilon \implies 2 - \frac{1}{n^2} > 2 - \epsilon$

Thus, we can make the elements of $S$ as close to 2 as desired, confirming that 2 is indeed the least upper bound.

Conclusion

Therefore, the supremum of the set $S$ is:

$\sup S = 2$

Questions and Additional Exploration

1. How does the behavior of $S$ change if we consider $S = \{ 2 - \frac{1}{n} : n \in \mathbb{N} \}$?
2. What would happen to the supremum if we changed the exponent from $n^2$ to $n^3$?
3. Can you construct a similar set $T$ such that $\sup T$ is different from $2$?
4. What is the infimum of the set $S$?
5. How does the concept of limits relate to the idea of supremum?

Tip

When analyzing sets defined by sequences, always consider both the behavior of the sequence as $n$ approaches infinity and the bounds imposed by the elements of the sequence.