Proving $(x_n,y_n)$ Converge to $(20,20) for All $k$

In summary, the given equations $y_0=k$, $x_{n+1}=30-\dfrac{y_n}{2}$, and $y_{n+1}=30-\dfrac{x_{n+1}}{2}$ will converge to $(20, 20)$ for all values of $k$. The dependence of $y_n$ on $x_n$ is not significant as the two variables are separable. By using the Banach fixed-point theorem and noting that $y_n$ is a contraction mapping, it can be proven that $y_n$ will converge to 20 regardless of the initial value of $k$. This, in turn, proves that $x_n$ will also converge to
  • #1
alexmahone
304
0
$y_0=k$ where $k$ is a constant.

$x_{n+1}=30-\dfrac{y_n}{2}$

$y_{n+1}=30-\dfrac{x_{n+1}}{2}$

Prove that $(x_n, y_n)$ converges to $(20, 20)$ for all values of $k$.

My attempt:

I wrote a computer program and verified this for a few values of $k$. But I don't know how to prove that $x_n$ and $y_n$ converge.
 
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  • #2
The dependence of $y_n$ on $x_n$ is a red herring, the two variables are separable as $x_{n + 1}$ can just be substituted:
$$y_{n + 1} = 30 - \frac{30 - \frac{y_n}{2}}{2}$$
Then once you find that $y_n$ converges to 20 regardless of $k$, it's easy to show that $x_n$ must too. You can easily find the limit of $y_n$ by noting that this is a contraction mapping and using the Banach fixed-point theorem ;)
 
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FAQ: Proving $(x_n,y_n)$ Converge to $(20,20) for All $k$

How do you prove that $(x_n,y_n)$ converges to $(20,20)$?

To prove that $(x_n,y_n)$ converges to $(20,20)$, you must show that for any given small positive number $\epsilon$, there exists an index $k$ such that for all indices $n \geq k$, the distance between $(x_n,y_n)$ and $(20,20)$ is less than $\epsilon$. This can be done using the definition of convergence and the properties of limits.

What is the definition of convergence for a sequence of points?

The definition of convergence for a sequence of points $(x_n,y_n)$ is that for any given small positive number $\epsilon$, there exists an index $k$ such that for all indices $n \geq k$, the distance between $(x_n,y_n)$ and a given point $(a,b)$ is less than $\epsilon$. This means that as $n$ becomes larger, the points in the sequence get closer and closer to the given point.

How do you find the value of $k$ to prove convergence?

To find the value of $k$ to prove convergence, you must use the definition of convergence and the properties of limits. You can start by setting $\epsilon$ to a small positive number and then use algebraic manipulation to solve for $k$. This value of $k$ will depend on the given sequence and the given point of convergence.

Can a sequence converge to multiple points simultaneously?

No, a sequence can only converge to one point at a time. This is because the definition of convergence states that for any given small positive number $\epsilon$, there exists an index $k$ such that for all indices $n \geq k$, the distance between the points in the sequence and the given point of convergence is less than $\epsilon$. If a sequence were to converge to multiple points, it would mean that the distance between the points in the sequence and each of the given points of convergence would have to be less than $\epsilon$ simultaneously, which is not possible.

Why is it important to prove convergence of a sequence?

It is important to prove convergence of a sequence because it allows us to understand the behavior of the sequence as $n$ becomes larger. It also allows us to make predictions about the values of the sequence at any given point. Additionally, proving convergence is often a necessary step in proving other important results in mathematics, such as the existence of limits and the continuity of functions.

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