Will we get all real numbers if we add all limits?

Question

Consider a set of all rational numbers from 0 to 1 inclusive.

If we add to this set all limits of all convergent sequences of these numbers, will we obtain a set of all real numbers from 0 to 1?

Answer 1

Yes. The rational numbers are dense in $[0,1]$. That is to say that every real number in $[0,1]$ is the limit of a rational sequence.

Answer 2

Yes. This is one construction of the real numbers: it is the completion of $\mathbb Q$ with respect to the metric $d(p,q) = |p-q|$.

Answer 3

Yes. In fact, one way to construct the real numbers is to union all of the rational numbers with all of their limits. Sequences of special importance are those which are Cauchy. A sequence $\{s_{i}\}$ is Cauchy if for every $\epsilon>0$ there exists an $N$ so that for all $m,n>N$, $|s_{n}-s_{m}|<\epsilon$. If we are in a metric space which has the property that every Cauchy sequence converges in the usual sense, then we say that the metric space is complete. The rationals are clearly not complete. Take the sequence $\{s_{n}\}=(1,1.4,1.41,1.414...)$. Then, clearly this sequence is Cauchy but does not converge. However, if we union the rationals with these limits, we have that every Cauchy sequence converges, so that the reals are complete. A key result in real analysis is that the reals are, in fact, complete. Here, it is important to note that there are other notions of completeness which are logically equivalent to that which I have already described.

Answer 4

Note that if $a\in [0,1]$ is an irrational number with decimal representation $$ a=0.a_1a_2a_3\cdots, $$ i.e. $a_i$ is the $i$th decimal of the number of $a$, then we can easily contruct a sequence of rational numbers converging to $a$ by taking $$ q_n=0.a_1\cdots a_n,\quad n\geq 1, $$ i.e. $q_n$ is the rational number matching the first $n$ decimals of $a$ (the rest of the decimals being zero). We see that $(q_n)_{n\geq 1}$ is a rational sequence converging to $a$.

Answer 5

Every real number has a decimal expansion, that is an infinite series expansion, which is a limit of rational numbers obtained by truncating the decimal expansion. That is the way most math students meet the limit concept first, even if the word 'limit' is not used. Assigning meaning to these infinite decimals to reveal the underlying properties they show has led to a formal definition of the reals, and also to a topological definition of limit and of compactness that is a tool used in essentially all non-discrete mathematics, such as real and complex analysis, number theory, probability, algebra, ...