Suppose that an is an additive sequence Show that the sequen

Suppose that (an) is an additive sequence. Show that the sequence ( an/n) converges.

Solution

This is not true. For example, consider the additive sequence (a_n) : {1,1,2.3,5,8,13,21,34,55,........}

Now, consider the sequence (a_n/n) : 1/1, 1/2, 2/3, 3/4, 5/5, 8/6, 13/7, 21/8,......

The sequence {1/1, 1/2, 2/3, 3/4, 5/5, 8/6, 13/7, 21/8, 34/9, 55/10,......} is clearly divergent. This is because, for any given N>0 (however large) you can always find an element in the sequence which exceeds it.