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ps8.lyx~
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ps8.lyx~
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_document
\begin_header
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\begin_body
\begin_layout Title
Problem Set 8
\end_layout
\begin_layout Author
Daryl Larsen
\end_layout
\begin_layout Section
PwCo
\begin_inset Formula $\implies$
\end_inset
UnCo
\end_layout
\begin_layout Standard
\series bold
(a)
\end_layout
\begin_layout Standard
\begin_inset Formula $\forall x\in[a,b],f_{n}(x)\rightarrow_{p}f(x),f(x)$
\end_inset
monotone increasing and continuous on
\begin_inset Formula $[a,b]$
\end_inset
.
Let
\begin_inset Formula $\delta>0$
\end_inset
such that
\begin_inset Formula $B(x,\delta)=(x+\delta,x-\delta)\cap[a,b]$
\end_inset
and let
\begin_inset Formula $y\in B(x,\delta)$
\end_inset
.
Because
\begin_inset Formula $f_{n}$
\end_inset
and
\begin_inset Formula $f$
\end_inset
are monotonically increasing, and by the triangle inequality,
\begin_inset Formula
\begin{align*}
\sup|f_{n}(y)-f(y)| & =|f_{n}(x+\delta)-f(x+\delta)+f_{n}(x-\delta)-f(x-\delta)|\\
& \leq|f_{n}(x+\delta)-f(x-\delta)|+|f_{n}(x-\delta)-f(x+\delta)|
\end{align*}
\end_inset
\end_layout
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