Thread: null sequences.

Prove or disprove:
If is a null sequence then one or both of is(are) null.

My attempt:
if both are null then certainly is null, no problem with that.

I assumed that is not null, is not null and as given, is null. I tried to bring a contradiction since i have a feeling that the statement in the question is true.

since its assumed that is not null so:
there exists such that for all there exists such that

similarly:
there exists such that for all there exists such that

and since is null:
there exists such that

now since i can't say that occur at the same i can't find a contradiction.

someone help.

Originally Posted by abhishekkgp

Prove or disprove:
If is a null sequence then one or both of is(are) null.

Assuming both converge, suppose that neither is a null sequence.
Then $\displaystyle (x_n)\to X\ne 0$ and $\displaystyle (y_n)\to Y\ne 0$.
What can you say about $\displaystyle (x_ny_n)\to~ ?$
What does that tell you.

Originally Posted by Plato

Assuming both converge, suppose that neither is a null sequence.
Then $\displaystyle (x_n)\to X\ne 0$ and $\displaystyle (y_n)\to Y\ne 0$.
What can you say about $\displaystyle (x_ny_n)\to~ ?$
What does that tell you.

probably and probably null sequences converge to 0 and hence a contradiction. But convergence has not yet been discussed in the book i am reading(first course in real analysis- stirling K berberian) so i can only guess.

Originally Posted by abhishekkgp

But convergence has not yet been discussed in the book i am reading(first course in real analysis- stirling K berberian) so i can only guess.

How very odd. I know Berberian's book on measure theory and find it odd.
Pray tell then what is a null sequence?

Originally Posted by Plato

How very odd. I know Berberian's book on measure theory and find it odd.
Pray tell then what is a null sequence?

a sequence in is said to be null if:
for all there exists such that

Originally Posted by abhishekkgp

a sequence in is said to be null if:
for all there exists such that

OK go back to the OP.
Let $\displaystyle \varepsilon = \frac{{\varepsilon _1 \varepsilon _2 }}{2}$.
Make $\displaystyle |x_ny_n|<\varepsilon$ then $\displaystyle \varepsilon _1 \varepsilon _2\le |x_n||y_n|=|x_ny_n|<\varepsilon$.
That is a contradiction.

Originally Posted by Plato

OK go back to the OP.
Let $\displaystyle \varepsilon = \frac{{\varepsilon _1 \varepsilon _2 }}{2}$.
Make $\displaystyle |x_ny_n|<\varepsilon$ then $\displaystyle \varepsilon _1 \varepsilon _2\le |x_n||y_n|=|x_ny_n|<\varepsilon$.
That is a contradiction.

the problem is that i can't( or yet not able to) say that there exists an such that . I can only say that and that's why i can't contradict anything.

Originally Posted by abhishekkgp

the problem is that i can't( or yet not able to) say that there exists an such that . I can only say that and that's why i can't contradict anything.

I think that I misread the OP.
Consider this example.
$\displaystyle x_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is even}} \\ {\frac{1}{n},} & {\text{ n is odd}} \\ \end{array} } \right.$ and $\displaystyle y_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is odd}} \\ {\frac{1}{n},} & {\text{ n is even}} \\ \end{array} } \right.$
What is $\displaystyle x_ny_n~?$

Originally Posted by Plato

I think that I misread the OP.
Consider this example.
$\displaystyle x_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is even}} \\ {\frac{1}{n},} & {\text{ n is odd}} \\ \end{array} } \right.$ and $\displaystyle y_n = \left\{ {\begin{array}{rl} {1,} & {\text{n is odd}} \\ {\frac{1}{n},} & {\text{ n is even}} \\ \end{array} } \right.$
What is $\displaystyle x_ny_n~?$

so it can happen that if we have $\displaystyle (x_n)$ not null, $\displaystyle (y_n)$ not null, we still have $\displaystyle (x_ny_n)$ as null!!

thank you for this.