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

[alexmahone]

$y_0=k$ 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.

 

[Nono713]

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 ;)