Answer to Question #18066 in Real Analysis for Matthew Lind

Question

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

Answer 1

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.