sets where sequence of continuous functions diverge

Related Exercise from Rudin’s Real and Complex Analysis.

Exercise 5.20

(a)
Does there exist a sequence of continuous positive functions $f_{n}$ on $\\mathbb{R}^{1}$ such that $\\{f_{n}(x)\\}$ is unbounded if and onlyif $x$ is rational?
(b)
Replace “rational” by irrational in (a) and answer the resultingquestion.
(c)
Replace “ $\\{f_{n}(x)\\}$ is unbounded”by “ $f_{n}(x)\\to\\infty$ as $n\\to\\infty$ ”and answer the resulting analogues of (a) and (b).

Solution:The answer to (a) is negative.This by showing thatthe subset of points where such sequence is unbounded must be $G_{\\delta}$ .But the rationals cannot be such,since in $\\mathbb{R}$ dense $G_{\\delta}$ sets must be of second category.

Rest of the answer not yet ready here

单词	SetsWhereSequenceOfContinuousFunctionsDiverge
释义	sets where sequence of continuous functions diverge Related Exercise from Rudin’s Real and Complex Analysis. Exercise 5.20 (a) Does there exist a sequence of continuous positive functions $f_{n}$ on $\\mathbb{R}^{1}$ such that $\\{f_{n}(x)\\}$ is unbounded if and onlyif $x$ is rational? (b) Replace “rational” by irrational in (a) and answer the resultingquestion. (c) Replace “ $\\{f_{n}(x)\\}$ is unbounded”by “ $f_{n}(x)\\to\\infty$ as $n\\to\\infty$ ”and answer the resulting analogues of (a) and (b). Solution:The answer to (a) is negative.This by showing thatthe subset of points where such sequence is unbounded must be $G_{\\delta}$ .But the rationals cannot be such,since in $\\mathbb{R}$ dense $G_{\\delta}$ sets must be of second category. Rest of the answer not yet ready here
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