# Answer to Question #18066 in Real Analysis for Matthew Lind

Question #18066

Show that if lim a_n = negative infinity, the lim 1/a_n = 0.

Expert's answer

Suppose lim a_n = - infinity.

We should prove that

lim 1/a_n = 0.

Fix any epsilon>0 and take A = 1/epsilon.

Since lim a_n = - infinity, there exists N such that forall n>N we have that a_n < -A

Therefore

|a_n| > A

|1/a_n| < 1/A =epsilon.

Thus for all n>N

|1/a_n| <epsilon.

This means that lim 1/a_n = 0.

We should prove that

lim 1/a_n = 0.

Fix any epsilon>0 and take A = 1/epsilon.

Since lim a_n = - infinity, there exists N such that forall n>N we have that a_n < -A

Therefore

|a_n| > A

|1/a_n| < 1/A =epsilon.

Thus for all n>N

|1/a_n| <epsilon.

This means that lim 1/a_n = 0.

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## Comments

Assignment Expert07.11.12, 18:48You're welcome. We are glad to be helpful.

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Matthew Lind07.11.12, 16:56Thank you very much.

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