[SOLVED] sum, difference and product of two null-sequence (need some help)

It's like this. (Thinking)

I didn't have any problems to prove some of definitions of sequences like :

Let

and

for any . Starting from some index every member of sequence are in -region of point x. Same, from some index every member of sequence are in -region of point y.

If we put then for every these inequality are true :

and

so from inequality of triangle for we have:

Because is fix real number and any small number so it's any small number so there we have that is:

true.

This and some another I got it OK (Wink)

But I can't, (Headbang)(Headbang)(Headbang) or don't see how to prove (or show) that sum of two null-sequences are again null-sequence (and difference of two sequences) becose it's to obviously. And for the product of two null-sequences is null-sequence and how we can relax conditions for the product of two null-sequences?

any help will be most appreciated (Thinking)

Thanks very much (Hi)(Bow)