  An elementary construction of the Wiener process is discussed, based on a
proper sequence of simple symmetric random walks that uniformly converge on
bounded intervals, with probability 1. This method is a simplification of F.B.
Knight's and P. R\'ev\'esz's. The same sequence is applied to give elementary
(Lebesgue-type) definitions of It\^o and Stratonovich sense stochastic
integrals and to prove the basic It\^o formula. The resulting approximating
sums converge with probability 1. As a by-product, new elementary proofs are
given for some properties of the Wiener process, like the almost sure
non-differentiability of the sample-functions. The purpose of using elementary
methods almost exclusively is twofold: first, to provide an introduction to
these topics for a wide audience; second, to create an approach well-suited for
generalization and for attacking otherwise hard problems.
