  For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka
< 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We
investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for
asymptotically stable random walks. This problem appeared first in the 1960's
in the analysis of a single-server queue when the traffic load tends to 1 and
since then is referred to as the heavy-traffic approximation problem. Kingman
and Prokhorov suggested two different approaches which were later followed by
many authors. We give two elementary proofs of our main result, using each of
these approaches. It turns out that the main technical difficulties in both
proofs are rather similar and may be resolved via a generalisation of the
Kolmogorov inequality to the case of an infinite variance. Such a
generalisation is also obtained in this note.
