The problem in the image gives a set $E = \left\{1 - \sum_{m=1}^{n} \frac{1}{4^m} : n \in \mathbb{N} \right\}$ and asks to determine certain properties of this set, such as its maximum, supremum, minimum, and infimum.

Let's analyze the set step by step:

The elements of the set $E$ are of the form: $E_n = 1 - \sum_{m=1}^{n} \frac{1}{4^m}$ This represents a sum of a geometric series with the common ratio $\frac{1}{4}$. We can compute the sum as follows:

$\sum_{m=1}^{n} \frac{1}{4^m} = \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \cdots \text{ up to } n \text{ terms}.$

The formula for the sum of a geometric series is: $S_n = \frac{\frac{1}{4} \left(1 - \left(\frac{1}{4}\right)^n\right)}{1 - \frac{1}{4}} = \frac{1}{3}\left(1 - \frac{1}{4^n}\right)$ Thus, each term of $E$ can be rewritten as: $E_n = 1 - \frac{1}{3}\left(1 - \frac{1}{4^n}\right) = 1 - \frac{1}{3} + \frac{1}{3 \cdot 4^n} = \frac{2}{3} + \frac{1}{3 \cdot 4^n}.$ Now, observe that as $n \to \infty$, the term $\frac{1}{3 \cdot 4^n} \to 0$, so the series converges to: $\lim_{n \to \infty} E_n = \frac{2}{3}.$

Thus, the infimum of $E$ is $\frac{2}{3}$. Since the sum approaches but never reaches this value, $\frac{2}{3}$ is the infimum but not the minimum.

The maximum occurs when $n = 1$, giving: $E_1 = 1 - \frac{1}{4} = \frac{3}{4}.$ Therefore, the maximum of the set is $\frac{3}{4}$.

The supremum is also $\frac{3}{4}$ because this is the largest element in the set.

Conclusion:

The correct option is (c):

• Maximum $E = \frac{3}{4}$,
• Supremum $E = \frac{3}{4}$,
• Minimum does not exist,
• Infimum $E = \frac{2}{3}$.

Would you like a detailed explanation of any step?

Here are 5 related questions:

1. How is the sum of a geometric series calculated?
2. What is the difference between the supremum and the maximum of a set?
3. Why does the set $E$ not have a minimum?
4. How can we determine the infimum of a set using limits?
5. What happens to the elements of $E$ as $n \to \infty$?

Tip: Always check whether a set contains its lower or upper bounds before deciding if it has a minimum or maximum!