There are a few possibilities, but here is the one approach. Even the starting point—the set of natural numbers $\mathbb{N}$—can be defined in several ways, but the standard definition takes $\mathbb{N}$ to be the set of finite von Neumann ordinals. Let us assume that we do have a set $\mathbb{N}$, a constant $0$, a unary operation $s$, and binary operations $+$ and $\cdot$ satisfying the axioms of second-order Peano arithmetic.
First, we need to construct the set of integers $\mathbb{Z}$. This we can do canonically as follows: we define $\mathbb{Z}$ to be the quotient of $\mathbb{N} \times \mathbb{N}$ by the equivalence relation
$$\langle a, b \rangle \sim \langle c, d \rangle \text{ if and only if } a + d = b + c$$
The intended interpretation is that the equivalence class of $\langle a, b \rangle$ represents the integer $a - b$. Arithmetic operations can be defined on $\mathbb{Z}$ in the obvious fashion:
$$\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$$
$$\langle a, b \rangle \cdot \langle c, d \rangle = \langle a c + b d, a d + b c \rangle$$
(Check that these respect the equivalence relation.)
Again, this is not the only way to construct $\mathbb{Z}$; we can give a second-order axiomatisation of the integers which is categorical (i.e. any two models are isomorphic). For example, we may replace the set $\mathbb{Z}$ by $\mathbb{N}$, since the two sets are in bijection; the only thing we have to be careful about is to distinguish between the arithmetic operations for $\mathbb{Z}$ and for $\mathbb{N}$. (In other words, $\mathbb{Z}$ is more than just the set of its elements; it is also equipped with operations making it into a ring.)
Next, we need to construct the set of rational numbers $\mathbb{Q}$. This we may do using equivalence relations as well: we can define $\mathbb{Q}$ to be the quotient of $\mathbb{Z} \times (\mathbb{Z} \setminus \{ 0 \})$ by the equivalence relation
$$\langle a, b \rangle \sim \langle c, d \rangle \text{ if and only if } a d = b c$$
The intended interpretation is that the equivalence class of $\langle a, b \rangle$ represents the fraction $a / b$. Arithmetic operations are defined by
$$\langle a, b \rangle + \langle c, d \rangle = \langle a d + b c, b d \rangle$$
$$\langle a, b \rangle \cdot \langle c, d \rangle = \langle a c, b d \rangle$$
And as before, we can give an axiomatisation of the rational numbers which is categorical.
Now we can construct the set of real numbers $\mathbb{R}$. I describe the construction of Dedekind cuts, which is probably the simplest. A Dedekind cut is a pair of sets of rational numbers $\langle L, R \rangle$, satisfying the following axioms:
- If $x < y$, and $y \in L$, then $x \in L$. ($L$ is a lower set.)
- If $x < y$, and $x \in R$, then $y \in R$. ($R$ is an upper set.)
- If $x \in L$, then there is a $y$ in $L$ greater than $x$. ($L$ is open above.)
- If $y \in R$, then there is an $x$ in $R$ less than $y$. ($R$ is open below.)
- If $x < y$, then either $x \in L$ or $y \in R$. (The pair $\langle L, R \rangle$ is located.)
- For all $x$, we do not have both $x \in L$ and $x \in R$. ($L$ and $R$ are disjoint.)
- Neither $L$ nor $R$ are empty. (So $L$ is bounded above by everything in $R$ and $R$ is bounded below by everything in $L$.)
The intended interpretation is that $\langle L, R \rangle$ is the real number $z$ such that $L = \{ x \in \mathbb{Q} : x < z \}$ and $R = \{ y \in \mathbb{Q} : z < y \}$. The set of real numbers is defined to be the set of all Dedekind cuts. (No quotients by equivalence relations!) Arithmetic operations are defined as follows:
- If $\langle L, R \rangle$ and $\langle L', R' \rangle$ are Dedekind cuts, their sum is defined to be $\langle L + L', R + R' \rangle$, where $L + L' = \{ x + x' : x \in L, x' \in L' \}$ and similarly for $R + R'$.
- The negative of $\langle L, R \rangle$ is defined to be $\langle -R, -L \rangle$, where $-L = \{ -x : x \in L \}$ and similarly for $-R$.
- If $\langle L, R \rangle$ and $\langle L', R' \rangle$ are Dedekind cuts, and $0 \notin R$ and $0 \notin R'$ (i.e. they both represent positive numbers), then their product is $\langle L \cdot L' , R \cdot R' \rangle$, where $L \cdot L' = \{ x \cdot x' : x \in L, x' \in L', x \ge 0, x' \ge 0 \} \cup \{ x \in \mathbb{Q} : x < 0 \}$ and $R \cdot R' = \{ y \cdot y' : y \in R, y \in R' \}$. We extend this to negative numbers by the usual laws: $(-z) \cdot z' = -(z \cdot z') = z \cdot -z'$ and $z \cdot z' = (-z) \cdot -z'$.
John Conway gives an alternative approach generalising the Dedekind cuts described above in his book On Numbers and Games. This eventually yields Conway's surreal numbers.
In fact, your teacher's definition doesn't seem to be right per Arthur's comment. I suspect that what they're trying to do is make the construction of the reals as simple as possible, but there is some unavoidable complexity there and - assuming you've copied what they've said correctly - they've stumbled on some of it.
Yes (with a bit of care), this is in fact one of$^1$ the standard ways of constructing the real numbers from the rational numbers.
We start with the notion of a Cauchy sequence of rational numbers only. Intuitively, these are the sequences which should converge but might not ... within $\mathbb{Q}$, anyways! We'd like to add a number corresponding to each such sequence. However, we have to be careful: different Cauchy sequences need not get different numbers! For example, consider $$(3,3.14,3.1415,3.141592,...)\quad\mbox{versus}\quad(3.1, 3.141, 3.14159, 3.1415926,...).$$ These each "point to $\pi$" (ignoring the minor fact that $\pi$ doesn't exist for us just yet!), but are different sequences.
So instead, we work with equivalence classes of Cauchy sequences of rationals. Specifically, there is a natural way - without "looking ahead" to $\mathbb{R}$! - to tell when two Cauchy sequences $(a_i)_{i\in\mathbb{N}},(b_i)_{i\in\mathbb{N}}$ "point to the same thing:"
$\lim_{i\rightarrow\infty}\vert a_i-b_i\vert=0$.
Writing "$\approx$" for the corresponding equivalence relation, we define $\mathbb{R}$ to be the set of $\approx$-classes of Cauchy sequences. Addition, multiplication, and so on of such classes can then be straightforwardly, if somewhat tediously, defined.
$^1$There are actually many different ways to "construct the reals." My personal favorite is via Dedekind cuts, FWIW. Given the plethora of options, this raises an interesting methodological concern: what exactly do we mean by "construct the real numbers," or even "the real numbers" for that matter? If I use a different construction than you, do we "disagree about $\mathbb{R}$?"
Addressing this concern satisfactorily turns out to be rather involved. I'm mentioning it here, however, since I think it is one which can reasonably occur early on and will only cause confusion if swept under the rug. So, even though it uses jargon which presumably you have not seen yet, I think it's worth stating the key theorem if only so that you know that such a thing exists:
There is exactly one complete real closed field up to isomorphism.
While the precise meaning of the above probably isn't clear, the general idea is quite simple: even if you and I think of $\mathbb{R}$ in terms of different constructions (e.g. Cauchy sequences vs. Dedekind cuts), "your version of $\mathbb{R}$" and "my version of $\mathbb{R}$" will be basically identical since they'll share a couple key mathematical properties. It turns out that there is a lot of subtlety here - see e.g. the discussion here - but ultimately the point is sound.
Best Answer
The idea is indeed correct, but needs a little work to formalise. You note that for every real $r$ there is a sequence $(a_n(r))_n$ of points that are all distinct and say (strictlty) increasing, so that $a(r)_n \rightarrow r$ as $n \rightarrow \infty$. You could choose the finite decimal approximations of $r$ (when we write $r$ as an infinite decimal expansion, which we can always do), if you want to be concrete.
Then the sets $A(r) = \{a_n(r): n \in \mathbb{N} \}$ would be almost disjoint for every pair of distinct $r_1 \neq r_2$; their intersection can be at most finite, or we would have a common subsequence tending to 2 limits, which cannot be. In particilar $r \rightarrow A(r)$ is an injection of $\mathbb{R}$ into $\mathscr{P}(\mathbb{Q})$.