Exercise 3.3.7

Question

Give an example to show that \href{./exercise_3-3-6}{Proposition 14.3.6} fails if the phrase "converges uniformly" is replaced by "converges pointwise".

Prakash Anand · Accepted Answer

Consider the same example as in \href{./exercise_3-2-2}{Exercise 3.2 .2(c)}. That is, $g^{(N)}=\sum_{n=1}^{N} x^{n}$. Each of these is a finite sum and hence bounded. However, this sequence converges pointwise to $g(x)=\frac{x}{1-x}$ which is not bounded on $(-1,1)$.

Exercise 3.3.7

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Exercise 3.3.7

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