So, I went to desmos website and drew two graphs. One was the graph of $y=x$ and another was the graph of $y=1$. The graphs came out to be something like this But, shouldn't the line y=1 coincide with y=x, because, it's the equation of the slope of y=x. Where am I going wrong?

I'm trying to perform an analysis of a recursion. I'll provide a bit of background and then the actual question later. Let $\left\{ \omega_j \right\}_{j\in\mathbb{N}} = \left\{ \arctan \left(2^{-j}\right) \right\}_{j\in\mathbb{N}}$, let $\theta \in \left[-\sum_{j=0}^{+\infty} \omega_j, \sum_{j=...

Here is a joke I invented (based on a famous one) and had mixed reaction. A young mathematician comes to present to a famous mathematician his conjecture and ideas. "You are absolutely wrong," the famous mathematician dismissed the young one. Next enters another young mathematician and presents ...

Let $\theta \in [0,2\pi]$, and let $\theta_j$ a sequence such that $\forall j \; \theta_j \in [0,2\pi]$, $\theta_j \to \theta$ with the property $$ |\theta_j - \theta| \geq |\theta_{j+1} - \theta|. $$ Let $R_{\psi}$ the rotation matrix, counter clockwise, respect to $\psi$. By drawing something...