Integers are easy. You start with 1 and successfully add or subtract 1. Rational numbers are just ratios of integers. But there clearly are other numbers like $\sqrt{2}$ which even the ancient greeks realized were not rational numbers, yet clearly were numbers, representing a very real distance. Dedekind was the first to try to define the totality of real numbers using sets, what today we call a $\textit{Dedekind cut}$:

Given a pair of sets $A$ and $B$ for which every element of $A$ is strictly less than every element of $B$ and for any positive distance $\epsilon > 0$ we can find an $a \in A$ and $b \in B$ such that $|b-a| < \epsilon$, then this pair of sets defines a real number. Furthermore, for every real number there is such a pair of sets.

In other words $\sqrt{2}$ is that unique number that is greater than every number less than $\sqrt{2}$ and less than every number greater than $\sqrt{2}$. Somehow saying it this way sounds less impressive. But the point is that whenever we can cut the real number line in this way, at the cut there must be an actual and unique real number.

Others soon joined this effort to define the real numbers including Karl Weierstrass, Georg Cantor, Eduard Heine, and H. Charles Méray. The most common approach was some version of equivalence classes of Cauchy sequences. In 1821 Cauchy considered the partial sums of an infinite series $a_1 + a_2 + a_3 + \cdots$. The partial sums are $S_1 = a_1, S_2 = a_1 + a_2, S_3 = a_1 + a_2 + a_3,$ and so on. Cauchy observed that if the series converges to a limit $S$, we can bring all of the partial sums arbitrarily close together just by going sufficiently far out this sequence. In formal language, given any positive quantity $\epsilon$ no matter how small, there is some some value $N$ beyond which all the partial sums are within $\epsilon$ of each other. This is not hard to prove. Convergence to $S$ means that beyond some value $N$ all of the partial sums are within $\epsilon/2$ of $S$, and thus within $\epsilon$ of each other. But Cauchy went further than this. He asserted that if the partial sums become arbitrarily close, then the series must converge. In his words, ``It is necessary and it suffices that, for infinitely large values of the number $n$, the sums $S_n, S_{n+1}, S_{n+2},\ldots$ differ from the limit $S$, among each other, by infinitely small quantities.''

He gave the proof I showed in the previous paragraph that this is necessarily the case. He skipped over the ``it suffices'' part. This is not an insignificant omission. Most tests of series convergence rely on proving that the partial sums will eventually be arbitrarily close. Dirichlet's increasing sequence that is bounded above is a classic example of a sequence that must be $\textit{Cauchy}$. The terms must be getting arbitrarily close together as they bump up against that upper limit. Formally, a sequence $S_1, S_2, S_3, \ldots$ is Cauchy if for any $\epsilon > 0$, there is some value $N$ beyond which all of the terms are within $\epsilon$ of each other. In symbols, $m,n \geq N \Rightarrow |S_m-S_n| < \epsilon$. We want them to converge to a real number, but what do we mean by the real number to which they converge? One can use a Cauchy sequence to define a Dedekind cut. The approach used by many others was to use the Cauchy sequences themselves to define real numbers.

Two Cauchy sequences are said to be equivalent if they come arbitrarily close together. In other words, $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ are equivalent if given any $\epsilon >0$ there is there some point $N$ beyond which the $a$'s and the $b$'s are within $\epsilon$ of each other. This can be used to define an equivalence class. One equivalence class consists of all the Cauchy sequences that are equivalent to each other. The real numbers are simply the equivalence classes of Cauchy sequences.

The definition of the real numbers as limits of Cauchy sequences clarifies what we mean when we say that $$\sqrt{2} =1.414213562373095\ldots.$$ That decimal expansion is actually a Cauchy sequence: $$ 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, \ldots$$ Note that each term of this sequence is a rational number, so its meaning is clear. The infinite decimal defines one of the equivalent Cauchy sequences that can be used to represent $\sqrt{2}$. Once we understand infinite decimals as Cauchy sequences, we see that most real numbers have a unique Cauchy sequence that is encoded by its decimal expansion. The exceptions are numbers of the form an integer multiplied by a power of 10 (a positive or negative power of 10). Thus $0.32 = 32\times 10^{-2}$ corresponds to two Cauchy sequences: $0.32000\ldots$ and $0.31999\ldots$. Note that since 1 is a power of 10, every integer corresponds to two Cauchy sequences. In particular, $1.0000\ldots = 0.9999\ldots$. These must be the same because their Cauchy sequences are equivalent. Forget about all the tricks you may have seen to ``prove'' that these are the same. Their equality is based on the definition of the real numbers.

I cannot help mentioning that there is another Cauchy sequence of rational numbers that has been known since ancient times to represent $\sqrt{2}$, $$ \frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \frac{41}{29}, \ldots.$$ The successor to $a/b$ is $(a+2b)/(a+b)$. These pairs, $a/b$, all satisfy $a^2-2b^2 = \pm 1$. To verify this, just observe that $1^2 - 2\cdot 1^2 = -1$ and \begin{eqnarray*} (a+2b)^2 - 2(a+b)^2 & = & a^2 + 4ab + 4b^2 -2(a^2+2ab+b^2) \\ & = & -(a^2 -2b^2). \end{eqnarray*} The values of $a$ and $b$ are clearly increasing. If $a^2-2b^2 = \pm 1$, then \begin{eqnarray*} (a-b\sqrt{2})(a+b\sqrt{2}) & = & \pm 1 \\ a-b\sqrt{2} & = & \frac{\pm1}{a+b\sqrt{2}} \\ \frac{a}{b} - \sqrt{2} & = & \frac{\pm 1}{b(a+b\sqrt{2})} \\ \left| \frac{a}{b} - \sqrt{2}\right| & < & \frac{1}{a(a+b)}. \end{eqnarray*} The fractions $a/b$ alternate less than and larger than $\sqrt{2}$ and are progressively closer to it.