Urgent Question: Convergens of series

Could somebody here please verify my proof?

Given the two sequences and which is defined as

1) show that if , for is true. Then for

2) show that if , for . Then for

Solution 1)

I choose an which for all m, implies that .

I then choose a random there is an N'' such that for all

from which is follows that

Then by the triangle inequality

there is an N'' such that for all , thus

Solution 2)

will tend to 0 by part (a).

I have let the two last tex modules stand because I cannot get them to work with "math", "/math"

Best regards