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If we are given two functions which are lower semicontinuous, f and g, which map from the metric space M to R (real numbers), then the sum f + g is also lower semicontinuous because:
for all alpha which belongs to the real numbers, there is an m which belongs to M so f(m) < alpha
and
for all beta which belongs to the real numbers, there is an m which belongs to M so f(m) < beta
and the sum
f(m) + g(m) < alpha + beta
and alpha + beta belongs to the real numbers.
correct?