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It comes to my attention through Lucien's answer that " represented as an infinite sum of a sequence of rational numbers" can be interpreted two ways. It can be simply $x = \sum q_n $ where each $q_n$ is a rational number. This is the way I interpreted it and it's this interpretation that the rest of this answer is based on.

Or it could be interpreted as $x = \sum $(*some nice rule that gives a rational number based on n*). The examples of the OP are of this type and have a predictive quality. We can use them to calculate the value of the real number. My interpretation has no predictive quality as to what the $q_n$ terms will be; just that there *are* a series of rational terms that will converge to the real irrational x.

By my interpretation, all irrationals can be so represented (answer below). By Lucien's interpretation, they can not. His/Her reason is there only countably many rules. I'm not sure of that, but I believe irrationals being uncountable make them "arbitrary" and unpredictable. But I'd have a very difficult time formalizing that.

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Short answer: That is the definition of a real number.

Long answer:

The fundamental thereom of analysis is that there is an ordered field that extends the rationals such that the field has the least lower bound property. We define the real numbers to be that field.

This means, by definition, every real number is the limit of a convergent sequence of rationals. By definition.

Infinite sums are the limit of finite sums. Hence every real can be written as an infinite sum of rationals. This equivalent to the definition of real number.

The proof of the fundamental thereom is kind of tedious and long. It's not hard but the point is you do the proof *before* the reals are defined and the definition comes out during the proof.

Longer answer:

Outline of Fund Theorem:

Step 1: Define a "cut" to be a set of rationals with properties:

i) a cut is not empty.
ii) if p is in a cut then every rational number less than p is in the cut
iii) for any p in the cut you can find a larger rational that is in the cut

So a cut could be all the rationals less than but not equal to 3. Or all rational numbers whose squares are less than 2. (The first is going to eventually be equivalent to 3, and the latter is going to eventually be equivalent to $\sqrt 2$

Step 2: Define a < b to mean the cut a is a subset of the cut b.

Step 3: Show that the set of all cuts, let's call it R~ has the least upper bound property.

Sheesh. This is where it gets abstract. The least upper bound property means every bounded set in a Universal Set (such as what the Reals will be once we define them) has a distinct limit that is in the universal set. Example: Q does *not* have the least upper bound property.

So we can have a set of cuts called A. It can be bounded above meaning the is a cut, b, such that all the cuts in A are subsets of b. (Remember "smaller" means "is a subset of"). The union of all the cuts in A is bigger or equal to all cuts in A. the union is a cut itself. The union it the smallest cut that is bigger than all the cuts in A. So the union is a least upper bound and R~ has the least upper bound property.

Step 4: Define cut a "+" cut b to be the cut that contains the sums of elements from a plus elements from b. Define 0~ to be the cut that contains all the negative numbers. This satisfies addition properties.

Step 5: More about field and additive and order properties than you'd care to think about.

Step 6-8: Show that R~ a field.

Step 9: Show the Q~ = all the cuts that are defined to be all points less than a rational number is equivalent to Q. so Q has an extension that is equivalent to R~. We call the R, the real numbers.

So..... So each real number is simply the limit of all the rational numbers in some cut. The cut provides sequences of rational numbers that converge to that real number.

So every real number is the limit of a convergent sequence of rational numbers.