What is the proof that the limit of a Cauchy sequence of integers is an integer?

[xaara test]

prove that limit of a cuachy sequence of integers is an integer

 

[statdad]

First think about what would have to be true for a sequence of integers to be a Cauchy sequence - then do some work and present it here.

 

[xaara test]

a sequence of integers which is cauchy is eventually constant. is the constant term the limit?

 

[statdad]

Try to show "a sequence of integers which is cauchy is eventually constant" and the answer for the limit will become apparent.

 

[blue_raver22]

I would approach this statement by first defining what a Cauchy sequence is. A Cauchy sequence is a sequence of real numbers that becomes arbitrarily close to each other as the sequence progresses. In other words, for any small positive number, there exists a point in the sequence where all subsequent terms are within that small distance of each other.

Now, to prove that the limit of a Cauchy sequence of integers is an integer, we can use the definition of a limit. According to the definition, the limit of a sequence is the value that the terms of the sequence approach as the index of the terms goes to infinity.

Since we are dealing with a sequence of integers, we can say that the limit of the sequence is an integer if the terms of the sequence become arbitrarily close to that integer as the index goes to infinity. In other words, for any small positive number, there exists a point in the sequence where all subsequent terms are within that small distance of the integer.

To prove this, we can use the fact that integers are closed under addition and multiplication. This means that the sum and product of any two integers will always result in an integer. Since the terms of the Cauchy sequence are integers, the subsequent terms will also be integers as they get closer to the limit.

Therefore, we can conclude that the limit of a Cauchy sequence of integers is an integer, as the terms of the sequence will always converge to an integer due to the closure property of integers. This proves that the statement is true and the limit of a Cauchy sequence of integers is an integer.