Question: ( x n → x ⟹ lim sup n → ∞ f ( x n ) ≤ f ( x ) ) ⟹ { x ∈ X | f ( x ) &lt; c } is open for any c ∈ R ?

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Question:   (      x    n    →  x    ⟹        lim sup          n      →      ∞        f  (      x    n    )  ≤  f  (  x  )  )    ⟹    {  x  ∈  X      |    f  (  x  )  &amp;lt;  c  }    is open for any    c  ∈  R  ?

Julianna Crawford · Accepted Answer

Step 1If       W    c    =  {  x  ∈  X  :  f  (  x  )  &amp;lt;  c  } is not open, for some   c  ∈      R   , then there exist a   ξ  ∈      W    c   i.e.   f  (  ξ  )  &amp;lt;  c , such that       B    ε    (  ξ  )  ⊄      W    c   , for every   ε  &amp;gt;  0 .In partiucular, for   ϵ  =      1    n   , there would exist an       ξ    n    ∈  X , such that   d  (      ξ    n    ,  ξ  )  &amp;lt;  1      /    n , and       ξ    n    ∉      W    c   i.e.   f  (      ξ    n    )  ≥  c . But clearly,       ξ    n    →  ξ , and due to the given condition, we would have that  c  ≤  lim sup  f  (      ξ    n    )  ≤  f  (  ξ  )  &amp;lt;  c  .Contradiction.

Theodore Dyer · Accepted Answer

Step 1To show   {  x  ∈  X  :  f  (  x  )  &amp;lt;  c  } is open, it is enough to show that   B  =  {  x  ∈  X  :  f  (  x  )  ≥  c  } is closed.Proof: Let   α  ∈      B    ′   i.e.   α is a limit point of B.To show   α  ∈  BSince   α  ∈      B    ′   ,   ∃  (      x    n    )  ∈  B such that       x    n    →  αThen by the hypothesis,   lim sup  f  (      x    n    )  ≤  f  (  α  )As   (      x    n    )  ∈  B ,   f  (      x    n    )  ≥  c  c  ≤  lim sup  f  (      x    n    )Then   c  ≤  lim sup  f  (      x    n    )  ≤  f  (  α  )Hence   f  (  α  )  ≥  c implies   α  ∈  BThus the set B closed.

Question: (x_nrightarrow x Rightarrow lim text(sup)_(n rightarrow infty)f(x_n)le f(x))Rightarrow{x in X|f(x)<c} is open for any c in R?

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