7.4 Sums and Products of Null Sequences

Scratchwork for : I want to find so
that

i.e.

This suggests that I take .

Scratchwork for : I want to find so that

Now , and I can make by making and . Hence I want and . This suggests that I take .

Proof: Let be null sequences, and let
. Define
by

Then for all ,

Hence, is a precision function for , and is a null sequence.

If then
is a null sequence. Suppose ,
and define
by

Then for all ,

Hence is a precision function for , and hence is a null sequence. Since it follows that is a null sequence.