Prove that f ( lim sup x → + ∞ x n )= lim sup x → + ∞ f ( x n )The problem is divided into two questions: x n is a real valued sequence.Prove that f ( lim sup n → + ∞ x n ) = lim sup n → + ∞ f ( x n ) given that f is continuous and increasing.What can we say when f is decreasing.

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Prove that   f  (      lim sup          x      →      +      ∞            x    n    )=       lim sup          x      →      +      ∞        f  (      x    n    )The problem is divided into two questions:      x    n   is a real valued sequence.Prove that   f  (      lim sup          n      →      +      ∞            x    n    ) =       lim sup          n      →      +      ∞        f  (      x    n    ) given that f is continuous and increasing.What can we say when f is decreasing.

ipa1rafd · Accepted Answer

For any sequence   (      y          n        ) let       y          n              +        =      sup          k      ≥      n            y          k      . Let   L  =  lim sup      x          n       be a real number. Since   L  =  lim      x          n              +       and f is continuous, we have  f  (  L  )  =  lim  f  (      x          n              +        )  .Now we show that   f  (      x          n              +        )  =  f  (      x          n            )          +       for each n. Since       x          n              +        ≥      x          k       for each   k  ≥  n and f is increasing, we have   f  (      x          n              +        )  ≥  f  (      x          n            )          +      . For any   ε  &amp;gt;  0 there exists   k  ≥  n such that       x          n              +        −  ε  &amp;lt;      x          k      . Again using the fact that f is increasing we get   f  (      x          n              +        −  ε  )  ≤  f  (      x          k        )  ≤  f  (      x          n            )          +      . Letting   ε  →  0 and using the continuity of f we get   f  (      x          n              +        )  ≤  f  (      x          n            )          +      If f is decreasing then (−f) is increasing and thus   (  −  f  )  (  lim sup      x          n        )  =  lim sup    (  −  f  )  (      x          n        )(xn). This reduces to   f  (  lim sup      x          n        )  =  lim inf  f  (      x          n        ) after simplification.

Prove that f(underset(x -> +oo)(lim sup x_n)) = underset(x -> +oo)(lim sup ) f(_xn)

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