  We note that if a sequence of real numbers converges to some limit, then the
sequence of the corresponding strings in the surreal $+,-$ sign expansion
representation converges, for a natural notion of string convergence, to the
string corresponding to the limit, modulo an infinitesimal. The corresponding
statement would be obviously false if we were considering, as strings, decimal
or binary representations, instead. The string limit of a possibly transfinite
sequence of surreal numbers is always defined and, when considering increasing
sequences of ordinals, corresponds to taking the supremum. A transfinite sum
can be defined using the string limit and this sum agrees with the
representation of a surreal number in Conway normal form.

Logic