Thread: 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 { } (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 there exists infinitely many integers n such that:

| – x| <
..... "

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 and given N there exists some n N such that | – x| < "

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

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 { } (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 there exists infinitely many integers n such that:

| – x| <
..... "

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 and given N there exists some n N such that | – x| < "

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 . Since there are infinite indexes that fulfill the first form there must be some that fulfills it, otherwise only those indexes s.t. would fulfill it , and there'd be a finite number of them...

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

Tonio