null sequences.

**abhishekkgp** · May 6th 2011, 11:37 AM

Prove or disprove:
If $(x_ny_n)$ is a null sequence then one or both of $(x_n), \, (y_n)$ is(are) null.

My attempt:
if $(x_n)\, and \, (y_n)$ both are null then certainly $(x_ny_n)$ is null, no problem with that.

I assumed that $(x_n)$ is not null, $(y_n)$ is not null and as given, $(x_ny_n)$ 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 $(x_n)$ is not null so:
there exists $\epsilon_1>0$ such that for all $N$ there exists $n \geq N$ such that $|x_n|>\epsilon_1$

similarly:
there exists $\epsilon_2>0$ such that for all $N$ there exists $n \geq N$ such that $|y_n|>\epsilon_2$

and since $(x_ny_n)$ is null:
there exists $N$ such that $n \geq N \Rightarrow |x_ny_n|<\epsilon_1 \epsilon_2$

now since i can't say that $|x_n|>\epsilon_1 \, and \, |y_n|>\epsilon_2$ occur at the same $n$ i can't find a contradiction.

someone help.

**Plato** · May 6th 2011, 11:42 AM

Originally Posted by abhishekkgp

Prove or disprove:
If $(x_ny_n)$ is a null sequence then one or both of $(x_n), \, (y_n)$ is(are) null.

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

**abhishekkgp** · May 6th 2011, 11:51 AM

Originally Posted by Plato

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

probably $(x_ny_n) \rightarrow XY \neq 0$ 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.

**Plato** · May 6th 2011, 12:01 PM

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?

**abhishekkgp** · May 6th 2011, 12:05 PM

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 $(x_n)$ in $\mathbb{R}$ is said to be null if:
for all $\epsilon>0$ there exists $N$ such that $n \geq N \Rightarrow |x_n|<\epsilon$

**Plato** · May 6th 2011, 12:13 PM

Originally Posted by abhishekkgp

a sequence $(x_n)$ in $\mathbb{R}$ is said to be null if:
for all $\epsilon>0$ there exists $N$ such that $n \geq N \Rightarrow |x_n|<\epsilon$

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

**abhishekkgp** · May 6th 2011, 12:20 PM

Originally Posted by Plato

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

the problem is that i can't( or yet not able to) say that there exists an $n$ such that $|x_n||y_n|> \epsilon_1 \epsilon_2$ . I can only say that $|x_{n_1}||y_{n_2}|>\epsilon_1 \epsilon_2$ and that's why i can't contradict anything.

**Plato** · May 6th 2011, 04:44 PM

Originally Posted by abhishekkgp

the problem is that i can't( or yet not able to) say that there exists an $n$ such that $|x_n||y_n|> \epsilon_1 \epsilon_2$ . I can only say that $|x_{n_1}||y_{n_2}|>\epsilon_1 \epsilon_2$ and that's why i can't contradict anything.

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

**abhishekkgp** · May 6th 2011, 10:58 PM

Originally Posted by Plato

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

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

thank you for this.

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