Thread: Proof of Limit Properties

Let {xn} as n tends to infinity be a sequence convergent to 0. Let {yn} as n tends to infinity be a bounded sequence. Show that:

lim as n tends to infinity(xnyn)=0

And is is this true for all sequences of {yn} as n tends to infinity or only those that are bounded? Can you provide a proof or counterexample for this

Originally Posted by amm345

Let {xn} as n tends to infinity be a sequence convergent to 0. Let {yn} as n tends to infinity be a bounded sequence. Show that:

lim as n tends to infinity(xnyn)=0

And is is this true for all sequences of {yn} as n tends to infinity or only those that are bounded? Can you provide a proof or counterexample for this

Here is a hint to get you started:

Since is bounded it is always less than some number lets call it M. i.e for all n

Now try to show that

I got that part, now I just need to figure out what happens when yn is unbounded.
Any hints?

Originally Posted by amm345

I got that part, now I just need to figure out what happens when yn is unbounded.
Any hints?

The bounded part is needed.



this is unbounded