# Sequence Convergence proof maybe?

• Oct 26th 2008, 02:21 PM
Caity
Sequence Convergence proof maybe?
a) Suppose that $\displaystyle {X_n}$ and $\displaystyle {Y_n}$ converge to the same point. Prove that $\displaystyle {X_n} - {Y_n} \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.

b) Show the converse is false.

Again as usual... some kind of explanation to help me understand this helps alot... :)
• Oct 26th 2008, 02:57 PM
whipflip15
For part a) i would use the $\displaystyle \epsilon-\delta$ definition of limits and it should follow directly. For part b) you just need to show a counter example. Maybe look at $\displaystyle y=x\mbox{ and }y^2-x^2=1.$
• Oct 27th 2008, 05:28 AM
Caity
ok that was how I started it... thanks for the info...
• Oct 27th 2008, 07:02 AM
Plato
Quote:

Originally Posted by Caity
a) Suppose that $\displaystyle {X_n}$ and $\displaystyle {Y_n}$ converge to the same point. Prove that $\displaystyle {X_n} - {Y_n} \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.
b) Show the converse is false.

Suppose that $\displaystyle L$ is the common limit of the two sequences.
If $\displaystyle \varepsilon > 0 \Rightarrow \quad \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {x_n - L} \right| < \frac{\varepsilon } {2}\,\& \,\left| {y_n - L} \right| < \frac{\varepsilon } {2}} \right]$.
Then observe that $\displaystyle \left| {x_n - y_n } \right| \leqslant \left| {x_n - L} \right| + \left| {L - y_n } \right|$.
• Oct 27th 2008, 11:01 AM
Caity
I finished part a). Thanks for the help. Now about part b). First what exactly would the converse be? I know its If X then Y becomes If Y then X. I was thinking maybe by showing something like subtraction is not commutative. Like if X - Y $\displaystyle \rightarrow$ 0 is not the same as Y - X $\displaystyle \rightarrow$0.
• Oct 27th 2008, 12:43 PM
Plato
The converse is: Suppose that $\displaystyle \left( {x_n - y_n } \right) \to 0$ the $\displaystyle \left( {x_n } \right)$ & $\displaystyle \left( {y_n } \right)$ have the same limit.
• Oct 28th 2008, 07:38 AM
Caity
Quote:

Originally Posted by Plato
The converse is: Suppose that $\displaystyle \left( {x_n - y_n } \right) \to 0$ the $\displaystyle \left( {x_n } \right)$ & $\displaystyle \left( {y_n } \right)$ have the same limit.

Well I tried this Plato... but it comes out true and I'm supposed to be proving it false... If the limit of $\displaystyle x_n$ and $\displaystyle y_n$ go to the same number say 1. Then we have 1 - 1 = 0 which shows it is true.
• Oct 28th 2008, 09:01 AM
Plato
Try this counterexample.
$\displaystyle x_n = \left( { - 1} \right)^n + \frac{1} {n}\,\& \,y_n = \left( { - 1} \right)^n - \frac{1} {n}\,$

Note: we are not given that the sequences converge, only that their difference is null.
• Oct 30th 2008, 11:51 AM
Moo
Quote:

Originally Posted by Plato
Note: we are not given that the sequences converge, only that their difference is null.

Is it possible to find a counterexample with the two converging ?
• Oct 30th 2008, 12:39 PM
Plato
Quote:

Originally Posted by Moo
Is it possible to find a counterexample with the two converging ?

The answer is no, the limit must be the same for both sequences.
Suppose that $\displaystyle \left( {x_n } \right) \to L,\quad \left( {y_n } \right) \to K\,\& \,\left| {\left( {x_n } \right) - \left( {y_n } \right)} \right| \to 0$.
That means that almost all the terms of $\displaystyle \left( {x_n } \right)$ are ‘close’ to $\displaystyle L$; almost all the terms of $\displaystyle \left( {y_n } \right)$ are ‘close’ to $\displaystyle K$; and almost all the terms of $\displaystyle \left( {x_n } \right) - \left( {y_n } \right)$ are ‘close’ to $\displaystyle 0$.

Suppose $\displaystyle L \not= K$ then $\displaystyle \varepsilon = \frac{{\left| {L - K} \right|}} {4} > 0$.
But how can almost all of the three sequences be with in an $\displaystyle \epsilon$-distance of its limit?