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:

   \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!