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Proof of Zero Test for Sequences

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Let (a_n) evaluated from n=M to infinity be a sequence of real numbers. Then the limit lim as n-->infinity of a_n exists and is equal to zero if and only if the limit lim as n-->infinity of the absolute value of a_n exists and is equal to zero.

Prove and answer if it is still true if we replace zero in the statement above by some other number.

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The solution provides proof of zero test for sequences.

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