- #1

Bptrhp

- 8

- 4

- Homework Statement
- Let ##(x_n)## and ##(y_n)## be Cauchy sequences in ##\mathbb{R}## such as ##x_n-y_n\rightarrow 0##. Prove that if exists ##K>0## such as ##|x_n|\leq K,\forall \,n\in\mathbb{N}##, then there exists ##n_0\in\mathbb{N}## such as ##|y_n|\leq K, \forall \,n>n_0## .

- Relevant Equations
- ##|x_n|\leq K,\forall \,n\in\mathbb{N}##

I've started by writing down the definitions, so we have

$$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$

$$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w $$

$$(y_n) \,\text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|y_m-y_n|<w $$

I tried using properties of the absolute value and the only vaguely useful result I got is ##|x_n-t_n|\leq C+|t_n|##. I can't see how to use this to prove the desired result.

Any hints? I appreciate any help!

$$x_n-y_n\rightarrow 0\, \Rightarrow \, \forall w>0, \exists \, n_w\in\mathbb{N}:n>n_{w}\,\Rightarrow\,|x_n-y_n|<w $$

$$(x_n)\, \text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|x_m-x_n|<w $$

$$(y_n) \,\text{is Cauchy} \, \Rightarrow \,\forall w>0, \exists \, n_0\in\mathbb{N}:m,n>n_{0}\,\Rightarrow\,|y_m-y_n|<w $$

I tried using properties of the absolute value and the only vaguely useful result I got is ##|x_n-t_n|\leq C+|t_n|##. I can't see how to use this to prove the desired result.

Any hints? I appreciate any help!