Computing $\bigcup_{n=1}^\infty C_n$ and $\bigcap_{n=1}^\infty C_n$ for $C_{n}=\{(x,y) \in \mathbb{R^2} : y^{n} \le x \}$ I am looking for some help to understand the following because I am feeling really lost.
Say $$C_{n}=\{(x,y) \in \mathbb{R^2} : y^{n} \le x \}$$
then how can we find
$$\bigcup_{n=1}^{\infty}C_{n}$$
and
$$\bigcap_{n=1}^{\infty}C_{n}$$
My thoughts were to start off by listing some such as 
$$C_{1}=\{(x,y) \in \mathbb{R^2} : y \le x \}$$
$$C_{2}=\{(x,y) \in \mathbb{R^2} : y^{2} \le x \}$$
$$C_{3}=\{(x,y) \in \mathbb{R^2} : y^{3} \le x \}$$
for example, and drawing out the graphs. I also solved for x in those cases to see if I could use that.
I am just quite lost, I am not sure the way to best approach this.
 A: Overview 
Here is a visualization using the GeoGebra software:

The purple curve and area is $y^{101} \le x$.
The blue curve and area is $y \le x$.
Union
First candidate for the union seems to be the union of the coloured areas here:

To get the two wanted sets precise one seems to need a good case distinction for each.


*

*First we have $C_1 = \{ (x,y) \in \mathbb{R}^2 \mid y \le x \}$, thus the line $y = x$ and all points below

*For $x \in I_1 = (-\infty, -1)$ we note that the sets $C_k$ with odd $k$ (shown is $C_3 = \{ (x,y) \in \mathbb{R}^2 \mid y^3 \le x \}$ in green) approach the half line $L_1 = \{ (x,-1) \mid x \in I_1 \}$ from below but do not reach $L_1$. That boundary $L_1$ is open. I used red dotted lines in the image above for such.

*For $x \in I_2 = (0,1)$ we note that the sets $C_k$ (all $k$) cover more and more of the area bounded by $L_2 = \{ (0,y) \mid y \in (0,1] \}$ and $L_3 = \{ (x,1) \mid x \in [0,1) \}$ and $y=x$ but do not reach $L_2$ or $L_3$.


Intersection
First candidate for the intersection seems to be the area bounded by the purple curves.

It is based on this overview:

Shown are $C_2 = \{ (x,y) \in \mathbb{R}^2 \mid y^2 \le x \}$ in cyan and
$C_4 = \{ (x,y) \in \mathbb{R}^2 \mid y^4 \le x \}$ in green. 
On $I_2 = (0,1)$ for positive $y$ the line $y=x$ is the boundary and included in the intersection. For negative $y$ on $I_2$ the line $y^2 = x$ is the boundary.
On $I_3 = (1, \infty)$ all points from $I_3 \times [-1,1]$ are part of the intersection. This area is included in all $C_k$, especially the $C_k$ with even $k$ approach it from outside.
A: The way you are approaching it is correct.


*

*Identify the graphs that matter

*Sketch in the regions of the graphs, which correspond to points in the set

*Take the union/intersection of those regions


The function in question here is $y^n$, for all $n\ge1$. This takes on three major different forms, which you'll discover if you graph $y$ to a few powers.
Another helpful tip about these functions is that they always pass through (1,1) and (0,0), no matter how fast they grow. 
