Q: What is meant by "The set of rational numbers is not closed when taking limits"? The set of rational numbers:
$$\mathbb{Q} = \bigg \{ \frac{m}{n} | \hspace{.2cm} m \in \mathbb{Z}, n \in \mathbb{N} \bigg \}$$
is described as being not closed when taking limits. I don't really understand this. What is meant by this statement?
The set of real numbers is presented and described with contains all limits of sequences of rational numbers. Again, I am really confused and don't understand what is meant here.
Can you please explain these two confusions of mine?

A: The first statement means that you might have a sequence of rational numbers which converge to a limit, but the limit is not rational.  E.g.,
here's a sequence of rational numbers:
$3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ....$
in which each term is the first few digits of $\pi.$   The limit of this sequence is $\pi$, which is not rational.
The second statement says that the real numbers don't have this "flaw." Every convergent sequence of rational numbers (or for that matter, real numbers) converges to a real number.  So the reals ARE closed under limits.

A: The property of being "closed under limits" is more formally referred to as completeness. To talk about what this means we need the definition of a Cauchy sequence:
Definition: A sequence $(a_n)_{n=1}^\infty$ of points in $\mathbb{R}$ (or $\mathbb{Q}$, or $\mathbb{Z}$, etc.) is a Cauchy sequence if for all $\varepsilon>0$, there exists a natural number $N$ (depending on $\varepsilon$) such that whenever $m,n\geq N$, $|a_n-a_m|<\varepsilon$.
Hence a Cauchy sequence consists of points which may not eventually get arbitrarily close to some point in your space, but do get arbitrarily close to each other. However, if you draw what a Cauchy sequence of real numbers should look like, you'll probably notice that the points that you draw do seem to actually approach some point! This is because the real numbers are complete: all Cauchy sequences are actually convergent sequences, i.e. every Cauchy sequence in $\mathbb{R}$ converges to some point in $\mathbb{R}$.
How is this different for the rationals? B. Goddard gave a great example; the Cauchy sequence of truncated decimal expansions for $\pi$ is convergent to $\pi$ in $\mathbb{R}$ (it has to be, since the real numbers are complete!), but cannot converge in $\mathbb{Q}$. If it did, then that would imply that $\pi\in\mathbb{Q}$.
We can think of points like $\pi$ as being "holes" in the rational numbers that the real numbers fill in. In fact, the real numbers are the completion of $\mathbb{Q}$; that is, the smallest complete space containing $\mathbb{Q}$. This is why the real numbers "contain the limits of all sequences of rational numbers" - they are in fact precisely constructed to fit this role.
