I'm TAing real analysis 1 this semester, and that course ought to give an answers to this. To name a few: 1. Every Cauchy-sequence converges, i.e. every sequen

I'm TAing real analysis 1 this semester, and that course ought to give an answers to this. To name a few: 1. Every Cauchy-sequence converges, i.e. every sequence that ought to converge does so. This implies the existence of roots, pi, e, sine and cosine functions defined by series representations. 2. The real numbers are the only archimedean ordered field which satisfies point 1. 3. If f is a continuous function on the interval [0,1] with f(0) < 0, f(1) > 0, then f has a zero in [0,1]. "If a function which can be drawn in one stroke crosses the x-Axis, at has a zero at the intersection point". As derivatives and integrals are typically defined using limits, having good results about the existence of limits is important theory wise. That said, I guess you could built up most of analysis 1 around the concept of a computable number, restricting yourself to computable functions. But I'm not sure that would make the subject easier at all.