We saw in a previous page that limits work nicely with respect to the sum of two sequences. We prove an analogous result for multiplication here.
Theorem.
Let (_{
)
} and (_{
)
} be convergent sequences
with limits
Proof.
Let
Subproof.
Let
Subproof.
Let
Let
Subproof.
Let
Subproof.
Assume
Then
So
It follows that
So
So
So
The next result is rather useful: it is a special case of the last result when one of the two sequences tends to zero. In this very special case, it doen't matter if the other sequence doesn't converge: it suffices that this other sequence is bounded. The proof is very similar.
Theorem.
Proof.
You should note particularly that this last result say something
special about the limit zero. There is no general result sating that
the limit of the product of a convergent sequence (_{
)
}
and a bounded sequence (_{
)
} exists. For a simple
counterexample, take _{
=
} to be the constant sequence
and for _{
} the bounded divergent sequence (-1)^{
}.
Then (_{
)
} diverges for