Consider two real numbers x = LIMn→∞ an, y = LIMn→∞ bn. Show that if x = y, then x −1 = y −1

To show that if \( x = y \), then \( x^{-1} = y^{-1} \), we need to analyze the problem step by step and use some basic properties of limits and inverse functions. Let’s proceed.

### Given:
1. \( x = \lim_{n \to \infty} a_n \)
2. \( y = \lim_{n \to \infty} b_n \)
3. \( x = y \)

We are asked to show that \( x^{-1} = y^{-1} \).

### Step-by-Step Proof:

1. **Equality of Limits**:
   Since \( x = y \), and both are defined as limits, we can express this as:
   \[
   \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n
   \]
   This implies that for sufficiently large \( n \), \( a_n \) and \( b_n \) will be arbitrarily close to each other, meaning:
   \[
   \forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, \ |a_n - b_n| < \epsilon
   \]
   So, \( a_n \) converges to \( b_n \) as \( n \to \infty \).

2. **Inverse of the Limits**:
   Now, we want to consider the inverse of \( x \) and \( y \), i.e., \( x^{-1} \) and \( y^{-1} \).

   From the properties of limits, if a sequence \( \{a_n\} \) converges to \( x \) and \( x \neq 0 \), then the sequence \( \{a_n^{-1}\} \) will converge to \( x^{-1} \). Similarly, since \( \lim_{n \to \infty} b_n = y \) and \( y \neq 0 \), the sequence \( \{b_n^{-1}\} \) will converge to \( y^{-1} \).

   Therefore:
   \[
   \lim_{n \to \infty} a_n^{-1} = x^{-1}
   \quad \text{and} \quad
   \lim_{n \to \infty} b_n^{-1} = y^{-1}
   \]

3. **Conclusion**:
   Since \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n \) and \( a_n \) converges to \( b_n \), by the property of limits:
   \[
   \lim_{n \to \infty} a_n^{-1} = \lim_{n \to \infty} b_n^{-1}
   \]
   Thus, \( x^{-1} = y^{-1} \).

### Important Condition:
This result holds as long as \( x \neq 0 \) and \( y \neq 0 \). If \( x = y = 0 \), the inverse is undefined, so the statement doesn't apply in that case.

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Would you like further clarification on any part of the explanation, or details on limit properties?

Here are 5 related questions you might find useful:
1. How do we rigorously define the inverse of a limit of a sequence?
2. What happens if one of the numbers is zero?
3. How does continuity of the inverse function play a role here?
4. Can we generalize this result to other operations beyond the inverse?
5. What are the necessary conditions for limits to preserve operations like multiplication and division?

**Tip**: When dealing with limits, always check for the possibility of division by zero before applying operations like inversion!

Explore a detailed proof showing that if the limits of two sequences x and y are equal, then their inverses are also equal. The problem uses concepts from limits, sequences, and inverse functions. Suitable for advanced high school or undergraduate students.

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