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Given this succession of functions I have proved the pointwise convergence to the null function.

For the uniform convergence I calculate $\lim_{x\rightarrow 0^+} f_n(x)=0$ and $\lim_{x\rightarrow 1^-} f_n(x)=1$ so can I say sup$f_n(x)\ge1$ in $(0,1)$ and there isn't uniform convergence?

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The sequence converges pointwise to zero.

So exists a sequence $x_m \to 1$ such that $f_n(x_m) \to 1$

Thus $$\sup_{x \in (0,1)}|f_n(x)|\geq \sup_{x \in \{x_1,x_2,...x_m,..\}}|f_n(x)| \geq 1$$

So indeed your idea is correct.

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