Let $0 < b < 1$ and consider the sequence of functions $(f_n)$ defined on $[0,b]$ by $f_n(x) = x^n$. Use the definition to show that $f_n$ converges to 0 uniformly on $[0,b]$. [Hint: $|x_n −0| = x_n ≤ b_n$.
2026-04-24 14:41:55.1777041715
Use the definition to show that $f_n$ converges to 0 uniformly
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You can guess that the limit function is $f(x)=0 , \forall x\in[0,b]$ . Take any $\epsilon >0$ . Then for every $n>\frac{\ln \epsilon}{b}$ you have $|f_n(x)-f|=|x^n|<b^n<\epsilon $ for all $x\in[a,b]$ . Hence proved .