Why is the Archimedian Property used?

[Singularity]

We are currently busy in calculus in proving limits of series ans sequences exist and so on. We use epsilon arguments to prove these things. Somewhere in the proof, the Archimedian Property is used i.e. for every real number x there exists a natural number k such that k > x. I don't understand why this is needed. Can someone please shed some light on this for me? Help me to underdtand this method of proving things. It doenst seem at all logical to me.

 

[matt grime]

Switch it round: given any $$\epsilon>0$$ (ie 1/x) $$\exists n \in \mathbb{N}$$ with $$1/n < \epsilon$$ so that the sequence 1/n tends to zero, and by comparison with this we can show lots of other things tend to zero. The proof is quite nice, I don't know if you've seen it - 1/n is decreasing and bounded below - it converges therefore to some limit x, then (1/n)^2 tends to x^2 and also, since it is a subsequence of the first sequence, it tends to x, thus x=x^2, ie x=0,1. Obvioulsy it isn't 1 (prove rigorously using the negation of the definition of convergence if you must). There is also the formulation that given any d>0, there is an integer n with nd>1

 

[M98Ranger]

The Archimedian Property is a fundamental concept in mathematics that plays a crucial role in proving the existence of limits of sequences and series. This property states that for any real number x, there exists a natural number k that is greater than x. This may seem like a simple and obvious statement, but it has powerful implications in calculus.

In order to prove the existence of limits of sequences and series, we often use the concept of epsilon arguments. These arguments involve choosing a small positive number, epsilon, and showing that for any value of epsilon, we can find a corresponding natural number k such that the difference between the sequence or series and its limit is less than epsilon. This is where the Archimedian Property comes into play.

By using the Archimedian Property, we can guarantee that there will always be a natural number k that is greater than any given real number x. This allows us to choose a specific value of k that will satisfy the epsilon condition, making our proof valid. Without the Archimedian Property, we would not be able to make this crucial step in our proof.

In essence, the Archimedian Property allows us to bridge the gap between real numbers and natural numbers, which are often used in mathematical proofs. It is a powerful tool that helps us prove the existence of limits and further our understanding of calculus. So while it may seem confusing or illogical at first, the Archimedian Property is an essential concept in mathematics and plays a key role in proving theorems and solving problems. I hope this helps to shed some light on this topic and helps you to better understand the method of proving things in calculus.