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

Alexander Lewis 2022-10-15 Answered
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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Answers (1)

ipa1rafd
Answered 2022-10-16 Author has 11 answers
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 > 0 there exists k n such that x n + ε < 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.
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