# Characterisation of a point of accumulation - Serge Lang - Undergraduate Analysis

• Apr 11th 2010, 12:07 AM
Bernhard
Characterisation of a point of accumulation - Serge Lang - Undergraduate Analysis
I am an amateur math hobbyist working through Serger Lang's book "Undergraduate Analysis":

On page 39 Lang writes the following:

"Let {$\displaystyle x_{n}$} (n = 1,2, ... ) be a sequence and x a number. We shall say that x is a point of accumulation of the sequence if given $\displaystyle \epsilon$ there exists infinitely many integers n such that:

|$\displaystyle x_{n}$ – x| < $\displaystyle \epsilon$
..... "

Lang then goes on to characterise a point of accumulation in a second way, namely as follows:

"In the definition of point of accumulation we could have said that given $\displaystyle \epsilon$ and given N there exists some n $\displaystyle \geq$ N such that | $\displaystyle x_{n}$ – x| < $\displaystyle \epsilon$ "

I have tried to prove that these two conceptualisations are the same but cannot form a convincing and exact proof,

I would really appreciate help in this matter.

Bernhard
• Apr 11th 2010, 01:06 AM
tonio
Quote:

Originally Posted by Bernhard
I am an amateur math hobbyist working through Serger Lang's book "Undergraduate Analysis":

On page 39 Lang writes the following:

"Let {$\displaystyle x_{n}$} (n = 1,2, ... ) be a sequence and x a number. We shall say that x is a point of accumulation of the sequence if given $\displaystyle \epsilon$ there exists infinitely many integers n such that:

|$\displaystyle x_{n}$ – x| < $\displaystyle \epsilon$
..... "

Lang then goes on to characterise a point of accumulation in a second way, namely as follows:

"In the definition of point of accumulation we could have said that given $\displaystyle \epsilon$ and given N there exists some n $\displaystyle \geq$ N such that | $\displaystyle x_{n}$ – x| < $\displaystyle \epsilon$ "

I have tried to prove that these two conceptualisations are the same but cannot form a convincing and exact proof,

I would really appreciate help in this matter.

Bernhard

Assume first form, and let be $\displaystyle N\in\mathbb{N}$ . Since there are infinite indexes that fulfill the first form there must be some $\displaystyle n>N$ that fulfills it, otherwise only those indexes s.t. $\displaystyle n\leq N$ would fulfill it , and there'd be a finite number of them...

Assume the second form...and now you try this by yourself.

Tonio