What can I conclude using the following theorem?(adsbygoogle = window.adsbygoogle || []).push({});

Let the functions [tex] u_n (x) [/tex] be continuous on the closed interval [tex] a \le x \le b[/tex] and let them converge uniformly on this interval to the limit function [tex] u(x) [/tex]. Then

[tex]

\int_a^b u (x) \, dx=\lim_{n \to \infty} \int_a^b u_n (x) \, dx

[/tex]

Can I conclude that if [itex] \int_a^b u (x) \, dx \neq \lim_{n \to \infty} \int_a^b u_n (x) \, dx [/itex], then the sequence of functions [itex] u_n (x) [/itex] isNOTuniformly convergent on the interval?

-Also what is the contrapositive of the above theorem? (I am confused on how to negate [itex] P [/itex]=" Let the functions [itex] u_n (x) [/itex] be continuous on the closed interval [itex] a \le x \le b[/itex] and let them converge uniformly on this interval to the limit function [itex] u(x) [/itex]." )

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# Uniform Convergence and integration

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