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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

1. Fix
Because for and
because for
Set
By the triangle inequality,
and
So ie
1. If , then , so it converges to 0. If , fix
Because for
So ie
1. 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
1. 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,
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