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 *}