Problem 202 (difficulty: 3/10)

Does \(\displaystyle a_n^2\to a^2\) imply that \(\displaystyle a_n\to a\)? And does \(\displaystyle a_n^3 \to a^3\) imply that \(\displaystyle a_n \to a\)?

Solution:

\(\displaystyle (-1)^n\not\to 1\). But for \(\displaystyle a=0\) we have \(\displaystyle \delta_{a_n}(\varepsilon):=\delta_{a_n^3} (\varepsilon^3)\), if \(\displaystyle a>0\), then \(\displaystyle |a_n- a|=\frac{|a_n^3-a^3|}{a_n^2+aa_n+a^2} \leq\frac{|a_n^3-a^3|}{3(a/2)^2}\) for \(\displaystyle n\) big enough.

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