For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

Question

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are all equilvalent to lower semicontinuity. I can't find a proof for this, can anyone supply one?

Many thanks, A.