  Two proofs of the Koml\'os-Major-Tusn\'ady embedding theorems, one for the
uniform empirical process and one for the simple symmetric random walk, are
given. More precisely, what are proved are the univariate coupling results
needed in the proofs, such as Tusn\'{a}dy's lemma. These proofs are
modifications of existing proof architectures, one combinatorial (the original
proof with many modifications, due to Cs\"{o}rg\~o, R\'{e}v\'{e}sz,
Bretagnolle, Massart, Dudley, Carter, Pollard etc.) and one analytical (due to
Sourav Chatterjee). There is one common idea to both proofs: we compare
binomial and hypergeometric distributions among themselves, rather than with
the Gaussian distribution. In the combinatorial approach, this involves
comparing Binomial(n,1/2) distribution with the Binomial(4n,1/2) distribution,
which mainly involves comparison between the corresponding binomial
coefficients. In the analytical approach, this reduces Chatterjee's method to
coupling nearest neighbour Markov chains on integers so that they stay close.
