I am not sure what the terms in the sequence are. If they are (0,1,0,1,...) for n terms, the sup norm for the difference is always 1, so they don't converge. If you drop a requirement that only finite combinations of basis elements are allowed, then I have no problem with the Schauder basis.For a topic that is specified to be not topological, there is sure a lot of topology involved To be quite frank, we don't seem to understand what your objective is, mathman.

The definition of Cauchy sequence (assuming at least some kind of metric present) states roughly that ##d(a_n,a_m)## eventually becomes arbitrarily small. Therefore, whether a sequence is Cauchy depends on given topology (is a given ##\varepsilon##-ball contained in a certain open set or not). A basis is just a way to express the sequence elements in terms of basis elements.

Furthermore, we have no control how the Schauder basis operates. The definition states that every element can be expressed as a uniquely determined infinite combination of basis elements, that's it. (For instance, you could have almost all scalars equal to zero). We're not allowed to make arbitrary combinations and declare them elements in the space.

On the other hand the problem seems kind of trivial. There is a representation ##\sum _{n\in\mathbb N} s_ne_n = (0,1,0,1,\ldots)##. By definition the series converges so the partial sums give us a Cauchy sequence.