Let's consider the sequences $\{x_n\}$ and $\{y_n\}$ defined by the recursive formulas:

$x_{n+1} = \frac{1}{2}(x_n + y_n)$ $y_{n+1} = \sqrt{x_n y_n}$ with the initial conditions $x_1 > 0$ and $y_1 > 0$.

We need to prove that both sequences converge to the same limit.

Step 1: Monotonicity

Let's first prove that $\{x_n\}$ is decreasing and $\{y_n\}$ is increasing.

1. Sequence $\{x_n\}$ is decreasing:

   Since $x_{n+1} = \frac{1}{2}(x_n + y_n)$, and by the arithmetic-geometric mean inequality (AM-GM inequality):

   $\frac{x_n + y_n}{2} \geq \sqrt{x_n y_n}$

   This implies:

   $x_{n+1} \geq y_{n+1}$

   Therefore, $x_n \geq x_{n+1}$ for all $n$, so $\{x_n\}$ is decreasing.

2. Sequence $\{y_n\}$ is increasing:

   Similarly, $y_{n+1} = \sqrt{x_n y_n}$, and by the same AM-GM inequality:

   $y_{n+1} = \sqrt{x_n y_n} \leq \frac{x_n + y_n}{2} = x_{n+1}$

   Thus, $y_n \leq y_{n+1}$ for all $n$, so $\{y_n\}$ is increasing.

Step 2: Boundedness

Next, we show that both sequences are bounded.

1. $\{x_n\}$ is bounded below:

   Since $x_n$ is decreasing and positive (as $x_1 > 0$), it is bounded below by $0$.

2. $\{y_n\}$ is bounded above:

   Since $y_n$ is increasing and $x_n \geq y_n$, the sequence $\{y_n\}$ is bounded above by $x_1$.

Step 3: Convergence

Since $\{x_n\}$ is decreasing and bounded below, it converges to some limit $L_x$.

Similarly, since $\{y_n\}$ is increasing and bounded above, it converges to some limit $L_y$.

Step 4: Common Limit

Now, we show that $L_x = L_y$. Taking limits in the recursive relations:

1. From the equation for $x_{n+1}$:

   $L_x = \frac{1}{2}(L_x + L_y)$

   Simplifying this:

   $2L_x = L_x + L_y$ $L_x = L_y$

   So, both sequences converge to the same limit, say $L$.

2. From the equation for $y_{n+1}$:

   $L = \sqrt{L \times L} = L$

Thus, both sequences converge to a common limit $L$.

Conclusion

The sequences $\{x_n\}$ and $\{y_n\}$ both converge to a common limit $L$.

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Would you like more details or have any other questions?

Related Questions:

1. What is the significance of the AM-GM inequality in this proof?
2. How can we determine the actual value of the common limit $L$?
3. Are there other methods to prove the convergence of these sequences?
4. What happens if $x_1$ and $y_1$ are not positive?
5. Can we generalize this approach to other types of recursive sequences?
6. How does the bounded monotonic sequence theorem apply here?
7. What if the initial conditions were such that $x_1 = y_1$?
8. How does the concept of fixed points relate to the convergence of sequences?

Tip:

Always check for monotonicity and boundedness when proving the convergence of sequences. These are powerful tools that simplify the proof process.