So far in this course you have seen several sequences that converge with proofs of their convergence. These include:
We want to build up our repertoire of such sequences, and we start by looking at addition.
Theorem.
Let (_{
)
} and (_{
)
} be convergent sequences
with limits
Proof.
Subproof.
Let
Subproof.
Let
Let
Subproof.
Let
Subproof.
Assume
Then
So
It follows that
So
So
So
This result enables us to write down the limit of sequences such as
1+
Proposition.
Let (_{
)
} be a convergent sequence
with limit
Proof.
We are given that
Subproof.
So
These two results show a similar theorem also holds for subtraction.
Theorem.
Let (_{
)
} and (_{
)
} be convergent sequences
with limits
Proof.