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The problem asks to find the **supremum** and **infimum** of two sets, \( A \) and \( B \), where:

- \( A = \{0.2, 0.22, 0.222, \dots \} \) is a set of decimal numbers whose digits are successively \( 2 \)'s, suggesting that the sequence approaches the value \( 0.\overline{2} \), i.e., a repeating decimal.
- \( B \) is the set of decimal fractions between 0 and 1 whose only digits are 0 and 1, which includes numbers like \( 0.0, 0.1, 0.01, 0.10, 0.11, 0.001, \dots \).

### Set \( A \):

- The set \( A \) is an increasing sequence of decimal numbers.
- The sequence starts with \( 0.2 \), then \( 0.22 \), \( 0.222 \), etc., and seems to approach the repeating decimal \( 0.\overline{2} \).
- Since \( A \) is an increasing sequence and converges to \( 0.\overline{2} \), the **supremum** of \( A \) is \( 0.\overline{2} \), which is equivalent to \( \frac{2}{9} \).
- The **infimum** of \( A \) is the smallest number in the set, which is \( 0.2 \).

### Set \( B \):

- Set \( B \) consists of all decimal fractions between 0 and 1 that use only the digits 0 and 1, forming numbers like \( 0.0, 0.1, 0.01, 0.10, 0.11 \), etc.
- The smallest number in set \( B \) is \( 0.0 \), and the largest number (the supremum) is \( 0.\overline{1} \), which is equivalent to \( 0.111\cdots \) or \( \frac{1}{9} \).
- Therefore, the **infimum** of \( B \) is \( 0 \), and the **supremum** is \( 0.\overline{1} \), or \( \frac{1}{9} \).

### Final Answer:

- **For set \( A \)**:
  - Infimum = \( 0.2 \)
  - Supremum = \( 0.\overline{2} = \frac{2}{9} \)

- **For set \( B \)**:
  - Infimum = \( 0 \)
  - Supremum = \( 0.\overline{1} = \frac{1}{9} \)

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Do you want more details or have any questions?

Here are 5 related questions you might find helpful:
1. How do we calculate the limit of a repeating decimal sequence?
2. What is the difference between a supremum and a maximum?
3. How can we prove that \( 0.\overline{2} = \frac{2}{9} \) and \( 0.\overline{1} = \frac{1}{9} \)?
4. What is the definition of an infimum in terms of sets of real numbers?
5. Can there be a maximum element in set \( B \)?

**Tip**: When working with repeating decimals, you can convert them into fractions using geometric series!

Find the supremum and the infimum of the sets A and B, where A = {0.2, 0.22, 0.222,...} and B is the set of decimal fractions between 0 and 1 whose only digits are zeros and ones.

Learn how to find the supremum and infimum of the sets A = {0.2, 0.22, 0.222,...} and B, where B consists of decimal fractions between 0 and 1 with only digits 0 and 1. This real analysis problem explores converging sequences, supremum, and infimum definitions.

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