Convergence and limits
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Suppose that f_k -> f uniformly on (0,1). Let 0 < x < 1. Suppose that lim f_k(t) = A_k for k=1,2,... Show that {A_k} converges and lim f(t) = LIM A_k.
That is show lim LIM f_k(t) = LIM lim f_k(t).
Where lim represents the limit as t approaches x and LIM represents the limit as k approaches infinity.
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Solution Summary
This provides an example of showing convergence and limits. Interchanging limits for uniform functions are examined.
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Please see the attachment.
Proof:
From the condition, we know that converges to uniformly on . For any , we have . We want ...
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