Complete Metric Spaces: Prove that the limit of a uniformly convergent sequence of functions is itself continuous.
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Complete Metric Spaces
Problem 1: Prove that the limit f(t) of a uniformly convergent sequence of functions {f_n(t)} continuous on [a,b] is itself a function continuous on [a,b].
Hint. Clearly
|f(t) - f(t_0)| < |f(t) - f_n(t)| + |f_n(t) - f_n(t_0)| + |f_n(t_0) - f(t_0)|,
where t, t_0 are real numbers of [a,b]. Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large n.
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