The proof of the above proposition is a direct verification based on
definitions.
The study of closeness of the partial minimum is based on the following
observation.
Suppose the level set
is nonempty for some
.
Let
be a sequence such that
.
Then
The set
is closed if
is closed. The set
is a projection of
.
Its closeness may be studied by means of the proposition
(
Preservation of closeness
result
). The intersection preserves the closeness. Hence, we arrive to the
following proposition.
