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{\bf Question}

Show that a function $f:{\bf R^n}\rightarrow R$ which is unbounded
above and below cannot be represented as a monotonic limit of
simple functions.


\vspace{0.25in}

{\bf Answer}

Let $\{f_n\}$ be an increasing sequence of simple monotonic
functions.

Let $\ds f_1=\sum_{i=1}^n c_iX_{Ei}$ and let $\ds
c=\min_{i=1\cdots n}c_i$

Then for all $x\epsilon{\bf R^n}$ and for all $n\epsilon{\bf N},
\,\,\, F_n(x)\geq c$

Hence if $f$ is unbounded below we cannot have $f_n\rightarrow f$
everywhere.  Similarly for decreasing sequences.


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