Computers are finite machines. The set of integers $\mathbb{Z}$ is
infinite and the set of reals $\mathbb{R}$ is much larger.
(The relative size of the reals is made precise by the notion of
caridnality.)
This is an issue.
Infinite requirements
Moreover, we need an infinite string of digits to represent a
single real number!
For every nonnegative real number $x$, there exists a natural
number $k$ and a sequence $d_n \in \{0, 1, \ldots, 9\}$ such that
\begin{align*}
x = k + \sum_{n=1}^{\infty} d_n(10)^{-n}.
\end{align*}
Either the sequence $d_n$ is unique, or there exists a sequence
ending in an infinite string of $0$'s and a sequence ending in
an infinite string of $9$'s.
This is another issue.
For every nonnegative real number $x$, there exists a natural
number $k$ and a sequence $d_n \in \{0, 1, \ldots, 9\}$ such that
\begin{align*}
x = k + \sum_{n=1}^{\infty} d_n(10)^{-n}.
\end{align*}
Either the sequence $d_n$ is unique, or there exists a sequence
ending in an infinite string of $0$'s and a sequence ending in
an infinite string of $9$'s.
For an accessible proof of this result, please refer to Section
$2.16$ of Kenneth Ross's
Elementary Analysis.