Dealing with the first one (lower semicontinuity), assume f is not bounded below, then there exists a sequence such that , but is compact, so it's sequentially compact, so without loss of generality we get . What can you say about .
Let f(x) be a real-valued function (an extended real valued function) on a metric space (M; d). Then
and
A function is called lower (upper) semicontinuous at a point y if
and
and
Let a metric space M be complete, and let K be a compact subset in M. Prove that a lower semicontinuous function on K is bounded from below and attains its minimal value and an upper semicontinuous function on K is bounded from above and attains its maximal value.
Any idea of how to approach this??