Exercise 3.3.5

Question

Give an example to show that \href{./exercise_3-3-4}{Proposition 3.3.4} fails if the phrase "converges uniformly" is replaced by "converges pointwise".

Prakash Anand · Accepted Answer

Take our good friend $f^{(n)}=x^{n}$ where $f^{(n)}:(-1,1) ightarrow \mathbb{R}$ and converges pointwise to the zero function $(f(x)=0$ for all $x)$. And take a sequence in $(-1,1)$ that converges to 1 . Then the theorem states that $f^{(n)}\left(x^{(n)}ight)$ converges to $f(1)=0$, but if we take the sequence $x^{(n)}=1-\frac{1}{n}$ we get that $f^{(n)}\left(x^{(n)}ight)=\left(1-\frac{1}{n}ight)^{n} ightarrow e^{-1} 
eq 0$.

Exercise 3.3.5

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Exercise 3.3.5

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