A sequence {a n } converges to a real number A iff for each real number > 0, the

Question

A sequence {an} converges to a real number A iff for each real number > 0, there exists a positive integer n* such that |an - A| < for all n n*.

The sequence is an = n / (n2 - 2). I need to prove that the limit as an approaches infinity is A = 0. Which means:

|n / (n2 - 2)|

= |n| / |(n2 - 2)|

= n / (n2 - 2), for all natural numbers n > 1

< (n + 1) / n, Because 0 < n < n + 1 for all n > 0 and 0 < n < n2 - 2 for all n > 2. So by the rule that if 0 < a < c and 0 < d < b then a/b < c/d.

= 1 + (1/n)

<

Now 1 + (1/n) < is true if n > 1 / ( - 1). So |an - 0| < if n 3. So we write n* > max{3, 1/(-1)}.

Explanation / Answer

This is correct until the very last line, where n* should be equal to that maximum, considering by the definition of the limit, there "exists" and n*, and so forth. Thus, define n* := max{3,1/(-1)}. Thereore, for all natural numbers greater than or equal to that n*, the magnitude is epsilon-small.

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