I am trying to find out what is wrong with the proof that all convergent sequence of functions is uniformly convergent and the limit function is continuos. It basically follows from these theorems:
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Every convergent sequence is Cauchy Sequence:[1]
All Cauchy sequence converges uniformly and its limit continuous[2],[3]
The proposition and theorem used:[4],[5]
I found a couterexample for a sequence of function $f_n$ $$f_n:[0,1]\rightarrow R$$ $$f_n=x^n$$ which converges to $f(x)=\left\{\begin{matrix}1(x=1) \\ 0 (0\leq x<1) \end{matrix}\right.$
which is not continuous