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The given problem asks to find the **supremum** and **infimum** of the set of numbers \( \frac{(n + m)^2}{2nm} \), where \( n, m \in \mathbb{N} \) (the set of natural numbers).

Let's rewrite the expression:

\[
S(n, m) = \frac{(n + m)^2}{2nm}
\]

### Step 1: Simplifying the expression
We can simplify \( S(n, m) \) by expanding the numerator:

\[
S(n, m) = \frac{n^2 + 2nm + m^2}{2nm} = \frac{n^2}{2nm} + \frac{2nm}{2nm} + \frac{m^2}{2nm}
\]

\[
S(n, m) = \frac{n}{2m} + 1 + \frac{m}{2n}
\]

### Step 2: Investigating behavior

- For \( n = m \), the expression becomes:

\[
S(n, n) = \frac{(2n)^2}{2n^2} = 2
\]

So, \( S(n, n) = 2 \).

- For large \( n \) and small \( m \), or vice versa, the terms \( \frac{n}{2m} \) or \( \frac{m}{2n} \) dominate, and the value of \( S(n, m) \) increases.

### Step 3: Infimum

Since both \( n \) and \( m \) are natural numbers, the smallest possible values of \( n \) and \( m \) are 1. Let's check the value at \( n = m = 1 \):

\[
S(1, 1) = 2
\]

Thus, the infimum of the set is \( \boxed{2} \).

### Step 4: Supremum

As \( n \) or \( m \) becomes large, the term \( \frac{n}{2m} + \frac{m}{2n} \) tends towards infinity. Therefore, the supremum of the set is \( \boxed{\infty} \).

### Conclusion:
- **Infimum** = 2
- **Supremum** = \( \infty \)

Would you like more details on the reasoning or any other aspect of this solution?

Here are 5 related questions:
1. How would the supremum change if the domain of \( n \) and \( m \) were extended to real numbers?
2. Can we determine the exact values of \( S(n, m) \) for specific pairs of \( n \) and \( m \)?
3. How does the expression behave for very large values of \( n \) compared to small values of \( m \)?
4. What does the supremum and infimum of a set tell us about its bounds?
5. What is the significance of finding the supremum and infimum in mathematical analysis?

**Tip:** Always simplify algebraic expressions to get better insight into the behavior of a function or sequence!

Find the supremum and the infimum of the set of numbers \( \frac{(n + m)^2}{2nm} \), where \( n, m \in \mathbb{N} \).

Explore how to determine the supremum and infimum for the function \( \frac{(n + m)^2}{2nm} \), involving algebraic manipulation and understanding of limits. This mathematical analysis is suitable for undergraduate students.

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