Exercise 6.7.9

Question

\begin{enumerate}[label=(\alph*)] 
    \item Find a counterexample which shows that WAT is not true if we replace the closed interval $[a, b]$ with the open interval $(a, b)$.
    \item What happens if we replace $[a, b]$ with the closed set $[a, \infty)$. Does the theorem still hold?
     \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
    \item $1/x$ over $(0,1)$, since $1/x$ is unbounded while any approximating polynomial must be bounded over $(0,1)$.
    \item $e^x$, since exponentials grow faster than polynomials, and therefore the difference between $e^x$ and any polynomial is unbounded over $[0, \infty)$.
     \end{enumerate}

Exercise 6.7.9

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Exercise 6.7.9

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