# Thread: Proof of Limit Properties

1. ## 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

2. 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 $\displaystyle y_n$ is bounded it is always less than some number lets call it M. i.e $\displaystyle |y_n|<M$ for all n

Now try to show that $\displaystyle |x_ny_n-0|< \epsilon$

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

4. 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.

$\displaystyle x_n=\frac{1}{n}, y_n=n^2$

$\displaystyle x_ny_n=n$ this is unbounded