When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a limited set of basic geometry operations to find relations between the elements, eliminate equations and then you'd eventually get the result. It seems like the kind of thing you could program a computer to do. So, I'm curious, does there exist any software to do this? I know there is lots of software for solving equations, but is there anything that lets you actually input a geometry problem without manually converting to equations? I'm not looking for anything too advance, even seeing just an attempt would be interesting. If there is anything decent, I think it'd be rather interesting to run the results on various competitions and see how many of the questions it solves.
The cardinality is at least that of the continuum because every real number corresponds to a constant function.  The cardinality is at most that of the continuum because the set of real continuous functions injects into the sequence space $\mathbb R^N$ by mapping each continuous function to its values on all the rational points. Since the rational points are dense, this determines the function.
The Schroeder-Bernstein theorem now implies the cardinality is precisely that of the continuum.
Note that  then the set of sequences of reals is also of the same cardinality as the reals.  This is because if we have a sequence of binary representations $.a_1a_2..., .b_1b_2..., .c_1c_2...$, we can splice them together via $.a_1 b_1 a_2 c_1 b_2 a_3...$ so that a sequence of reals can be encoded by one real number.
