Please consider the example below:
Although I understand much of the solution, I do not understand why $f_n(t) = \dfrac{1}{\sqrt t}~\forall~n \ge p$. Could someone help me with understanding this. Thank you!
2026-03-25 10:58:39.1774436319
Integral Metric on $C([0,1])$
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It is just by definition of $f_n$. It is defined for $t\in\big[\frac{1}{n^2},1\big]$ as $$ f_n(t)=\frac{1}{\sqrt{t}}.\tag{1} $$ If $n\ge p$ then $\big[\frac{1}{p^2},1\big]\subset\big[\frac{1}{n^2},1\big]$, hence, (1) holds for $n\ge p$ and $t\in\big[\frac{1}{p^2},1\big]$.