  This paper is concerned with an inverse obstacle scattering problem of an
acoustic wave for a single incident plane wave and a wave number. The
Colton-Sleeman theorem states the unique recovery of sound-soft obstacles with
a smooth boundary from the far-field pattern of the scattered wave for a single
incident plane wave at a fixed wave number. The wave number has a bound given
by the first Dirichlet eigenvalue of the negative Laplacian in an open ball
that contains the obstacles. In this paper, another proof of the Colton-Sleeman
theorem that works also for the case when we have a known {\it unbounded} set
that contains obstacles is given. Unlike original one, the proof given here is
not based on the monotonicity of the first Dirichlet eigenvalue of the negative
Laplacian. Instead, it relies on a {\it positive supersolution} of the
Helmholtz equation in a known domain that contains obstacles. Some corollaries
which are new and not covered by the Colton-Sleeman Theorem are also given.
