Cardinality of any bounded interval of ℝ = |ℝ|?

A bit of background on why I'm asking this:

Take the sequence of ℤ+, which has the same cardinality as ℤ:

1, 2, 3, 4, 5, 6, ...

Suppose we create a number in base 11 using x as the extra symbol, by concatenating the numbers, and delimiting with x.

1x2x3x4x5x6x...

This is a perfectly legal number in our base 11 scheme.

Now scale it:

0.1x2x3x4x5x6x...

This would be a real number between 0 and 1 [0, 1).

In other words, this real number represents the sequence ℤ+ and consists of a single point p on the real number line, 0 ≤ p < 1.

Take any other sequence in ℤ+, transform it the same way, and it can be represented as a point p on the real number line, 0 ≤ p < 1:

0.10x20x30x40x... times ten
0.1x4x9x16x25x... squares
0.41x8x20x9x5x... some random sequence

All finite sequences of similar length can also be represented similarly. For example, all finite sequences of length 2: {1, 1} {1, 2} {1, 3} ... can be ordered and represented by a single number in ℝ:

0.1x1x1x2x1x3x...

So we have the set of all sequences, finite and infinite, representable as a point p, 0 ≤ p < 1.

These sequences can be scaled to any arbitrary [q, r) such that rq, but always r > q.

Given that [q, r) is an arbitrarily small but bounded interval in ℝ, and that ℤ+ is represented as a single point p', qp' < r, can it be argued that the size of this set is larger than ℤ but smaller that ℝ?

I guess it all hinges on whether the cardinality of any bounded interval of ℝ = |ℝ|, hence the title of the question.

• Consider the function $\tan x$, which maps from a finite range to the entire set of real numbers. Note that $\tan x$ over its finite domain is one to one and onto and thus proves the cardinalities are equal. – abiessu Sep 11 '13 at 0:51
• The interval $[q,r)$ is neither infinitely small nor finite, for any $r > q$. The intersection or "limit set" of all these sets is just the single point $q$, but you can't map all the integer sequences to it. This is basically another example of behavior of a sequence not matching the behavior of the limit. – MartianInvader Sep 11 '13 at 0:54
• @abiessu, thank you for such a clear and concise answer. – user94213 Sep 11 '13 at 1:04
• Generally, the question has been asked ad nauseum on this site. math.stackexchange.com/questions/200180/… is a particular example, and there are more - much more - examples for closed, open, left-closed and right-open, and right-closed and left-open, and so on. – Asaf Karagila Sep 11 '13 at 6:00
• @Asaf Karagila - If the question is asked ad nauseum then that might be an indication that the matching done by SO isn't working so well. It was only when I revised the title to use "bounded interval" did proper matches show up. – user94213 Sep 11 '13 at 13:30

The function $\tan x$ (among others, but tangent is easy to use) can be mapped from any specific range in the real numbers to the entire set of reals. In particular, let the range $[a,b]$ be given where $a\ne b$, then $\forall x\in (a,b),$ $$f(x)=\tan\left({\pi\cdot (x-a)\over b-a}-{\pi\over 2}\right)$$ supplies this mapping.
Since it is possible to pick any real number $r$ and find $x_r$ such that $f(x_r)=r$, and vice versa, this function is one to one and onto, and is thus a complete mapping AKA bijection from a bounded interval of the reals to the entire set, which means that the cardinalities are the same.