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If f_n : A\rightarrow R sequnce of continuous functions converges uniformly to f prove that f is continuous
My work
Given \epsilon > 0
fix c\in A want f is continuous at c
|f(x) - f(c) | = |f(x) - f_n(x) + f_n(x) - f(c) | \leq |f(x) - f_n(x) | + |f_n(x) - f(c) |
the first absolute value less that epsilon since f_n converges uniformly to f
and since
f_n(x) is continuous at c so there exist \delta such that |x - c| < \delta
then |f_n(x) - f(c) | < \epsilon
Am i right ?
My work
Given \epsilon > 0
fix c\in A want f is continuous at c
|f(x) - f(c) | = |f(x) - f_n(x) + f_n(x) - f(c) | \leq |f(x) - f_n(x) | + |f_n(x) - f(c) |
the first absolute value less that epsilon since f_n converges uniformly to f
and since
f_n(x) is continuous at c so there exist \delta such that |x - c| < \delta
then |f_n(x) - f(c) | < \epsilon
Am i right ?