# fa.functional analysis – prove that the \$ D \$ set of \$ X \$ is closed

Let & # 39; s $$y _ {0} in C _ {0}$$ and define
begin {align *} D = left {x in C _ {0}: ; lim _ {n rightarrow infty} sup Green x – x _ {n} Green leqslant lim _ {n rightarrow infty} sup Green y _ {0} – x _ {n} Green = s right } mbox {.} end {align *}
Let $$(w _ {n})$$ a sequence such as $$w _ {m} rightarrow w$$, or $$w in overline {D}$$then $$Green w _ {m} – x _ {k} Green rightarrow Green w – x _ {k} Green$$. however
begin {align *} w Green w Green green u left green lim {rightarrow infty} (w _ {m} – x _ {n}) right Green = lim sup left ( lim _ {m rightarrow infty} Green w _ {m} – x _ {n} Green right) \ & = lim _ {m rightarrow infty} left ( lim sup Green w w {{}} – x _ {n} Green right) \ & leqslant lim _ {m rightarrow infty} s = s mbox {.} end {align *}
So $$w in D$$, concluding that $$D$$ is closed. My doubt is: why
begin {align *} lim sup left ( lim _ {m rightarrow infty} Green w _ {m} – x _ {n} Green right) = lim _ {m rightarrow infty} left ( lim sup Green w w {{}} – x _ {n} Green right)? end {align *}