The **algebra of limits** is a set of rules for how limits may be manipulated with other operators.

## For real sequences

For real convergent sequences and where and , and for a real number :

If also and

### Proof

- Fix

Because for and

because for

Set

By the triangle inequality,

and

So ie

- If , then , so it converges to 0. If , fix

Because for

So ie

- Fix As is convergent, it is bounded by some ie

Because for and

because for

Set

By the triangle inequality, and boundedness of ,

And

So ie

- By part 3,

Fix

Because for

Also, for

By the triangle inequality,

And for so

Now

And now

Setting we have

or in other words

Therefore,