Since nobody mentioned it yet, how about proving convergence and finding limits of some recursively defined sequences of real numbers? A few not too hard examples are:
1) $x_1=a, \ x_2=b, \ x_n=\frac{x_{n-1}+x_{n-2}}{2}$;
2) $x_0 >0, \ x_{n+1}=\frac{1}{2}(x_n+\frac{1}{x_n})$;
3) $x_0 = \sqrt{2}, \ x_{n+1}=\sqrt{2+x_n}$.
This gives you something instructive to do even before you discuss differentiation and integration. In my experience, most students figured out how to compute these limits; proving convergence was a harder sell, but in cases 2) and 3) it is accessible even to an average student.
