Sequence proof

**flipperpk** · October 5th 2009, 03:30 PM

Let $(a_{n})$ be a sequence which converges to 0, and let $(b_{n})$ be a sequence which is bounded but not necessarily convergent.

Prove that $lim (a_{n})(b_{n}) = 0$

No idea what to do here.

**hjortur** · October 5th 2009, 04:30 PM

Well I am not so sure about this but cant you do something in this direction?

If $b_n$ is bounded then there exists M such that
$ |b_n|\leq M $
For all $n\in \mathbb{N}$

Now then it is clear that:
$ \lim\limits_{n\to\infty}(a_n)(-M) \leq \lim\limits_{n\to\infty}(a_n)(b_n) \leq \lim\limits_{n\to\infty}(a_n)M $

$ 0\leq\lim\limits_{n\to\infty}(a_n)(b_n)\leq 0 $

Something like this?

**flipperpk** · October 5th 2009, 04:47 PM

I figured the squeeze theorem would show up in this one. I am horrible at generalizing like that. Thanks

**hjortur** · October 5th 2009, 04:59 PM

Not a problem, I am not so good either.

Thread: Sequence proof

LinkBack

Thread Tools

Display

Sequence proof

Similar Math Help Forum Discussions

Sequence proof

Sequence proof

Sequence Proof

Sequence proof

Sequence Proof

Search Tags