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.
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.
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.