It's like this.

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

But I can't,

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.