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How will I know if limits such as $|a_n-L|\Longleftrightarrow|a_n^2-L^2|$ as $a_n \rightarrow L$ are true?

In this case, I know that $|a_n^2-L^2| \Longrightarrow |a_n-L|$ as $a_n \rightarrow L$ is false because $L$ could be negative or that $a_n = L, -L, L, -L, ...$, but how do I determine if $|a_n-L| \Longrightarrow |a_n^2-L^2|$ is true?

Thank you!

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    Okay, thanks! I will edit the question2011-10-06

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If $a_n-L \to 0$, then $a_n \to L$; hence $a_n^2 - L^2 = (a_n - L)(a_n + L) \to 0 \cdot (L+L) = 0$.

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    My apologies, I was just wondering if $[a_n \rightarrow L] \Longleftrightarrow [a_n + 1 \rightarrow L + 1] \Longleftrightarrow [2a_n \rightarrow 2L]$ is indeed true.2011-10-07