"Everyone knows" that to show that $\mathbb{R}$ is uncountably infinite, it suffices to show that the real numbers in the interval $(0,1)$ cannot be listed, which can be accomplished by Cantor's famous diagonal argument.
My question is: How may I prove that the cardinalities of these two sets are the same? Is it only possible to show this by finding a bijection from one of these sets into another? Are there other ways?
On
By definition, showing two sets have the same cardinality means showing there exists a bijection between them.
To this end, $\arctan x$ can be used. That is, $\arctan:\Bbb R\to (-\frac {\pi}2,\frac {\pi}2)$ is a bijection.
So we just need a bijection between $(0,1)$ and $(-\frac {\pi}2,\frac {\pi}2)$. But this is easy. Let $c(t)=(1-t)(-\frac {\pi}2)+t(\frac {\pi}2)$.
Basically, you find a bijection between them to show both cardinalities are same. But here, even if you don't know the explicit map (bijection), between $\Bbb R$ and $(0,1)$, we can still say both are equivalent via some geometrical notion. For example, see the below figure
The idea is : (replace $-1$ by $0$ in figure) Imagine $(0,1)$ bent into a semicircle that rests on the number line at $O$. Rays from the center of the semicircle establish a one-to-one correspondence between points of $(0, 1)$ and points of the line.