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For each $n \ge 1$, let $f_n$ be a monotonic increasing real valued function on $[0, 1]$ such that the sequence of functions $\{f_n\}$ converges pointwise to the function $f \equiv 0$. Pick out the true statements from the following: a. $f_n$ converges to $f$ uniformly. b. If the functions $f_n$ are also non-negative, then $f_n$ must be continuous for sufficiently large $n$. |
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For (a) you know that $f_n(0)\to 0 $ and $f_n(1)\to 0$. That is: For given $\varepsilon>0$ there is $n_0\in\mathbb N$ such that $n>n_0$ implies $|f_n(0)|<\varepsilon$ and there is $n_1\in\mathbb N$ such that $n>n_1$ implies $|f_n(0)|<\varepsilon$. Now verify that for all $n>\max\{n_0,n_1\}$ you have $|f_n(x)|<\varepsilon$ for all $x\in [0,1]$. for (b) consider $f_n(x)=\begin{cases}\frac2nx &\mathrm{if\ }x>1-\frac1n\\\frac1nx&\mathrm{otherwise}\end{cases}$. (These are even strictly increasing.) |
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