To prove that if $x = y$, then $\frac{1}{x} = \frac{1}{y}$, we can proceed step by step, using the definition of limits and some properties of real numbers.

Given Information:

• $x = \lim_{n \to \infty} a_n$
• $y = \lim_{n \to \infty} b_n$
• We are given that $x = y$.

Objective:

Show that $\frac{1}{x} = \frac{1}{y}$, assuming that $x$ and $y$ are non-zero real numbers (since division by zero is undefined).

Proof:

1. Assumption: We are given that $x = y$. Therefore, by definition, the limits of the two sequences $a_n$ and $b_n$ are equal: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n.$ This implies that for large $n$, the sequences $a_n$ and $b_n$ become arbitrarily close to each other.

2. Reciprocals of Limits: Since $a_n \to x$ and $b_n \to y$, for sufficiently large $n$, the terms $a_n$ and $b_n$ are close to $x$ and $y$, respectively. Moreover, because $x = y \neq 0$, the reciprocals $\frac{1}{a_n}$ and $\frac{1}{b_n}$ also approach $\frac{1}{x}$ and $\frac{1}{y}$, respectively.

3. Reciprocal of Limits Property: It is a standard result in real analysis that if $\lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n$, and both limits are non-zero, then: $\lim_{n \to \infty} \frac{1}{a_n} = \lim_{n \to \infty} \frac{1}{b_n}.$ This is because if $a_n$ and $b_n$ approach the same non-zero limit, their reciprocals also approach the same limit.

4. Conclusion: Since $x = y$ and both are non-zero, it follows that: $\frac{1}{x} = \frac{1}{y}.$

Thus, we have shown that if $x = y$, then $\frac{1}{x} = \frac{1}{y}$.

Would you like further details or explanations?

Here are five related questions to expand on this topic:

1. What happens if $x = 0$ in the context of the limit?
2. How does the concept of limits apply to functions, not just sequences?
3. Can this proof be generalized to complex numbers?
4. What are some common errors in handling limits involving infinity?
5. How do the properties of limits and continuity relate?

Tip: When working with limits and reciprocal functions, always ensure the denominator doesn't approach zero, as division by zero is undefined.