(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider sequence of real numbers.

Theorem: If a= sup S, then there exist a sequence x_{n}E S such that x_{n}->a

Proof:

Take ε = 1/n and find x_{n}E S such that 0 ≤ a - x_{n}< 1/n.

Now show x_{n}-> a.

======================================

I am very very confused about this proof.

1) Why are they taking ε = 1/n? What motivates this?

2) It seems to me that n is simply a "subscript" of the sequence x_{n}and it's a bit weird to talk about ε = 1/n. Is there any relationship between the "n" in ε = 1/n and the "n" in the sequence x_{n}? Are they the SAME "n"?

3) In the proof, they say "find x_{n}E S such that 0 ≤ a - x_{n}< 1/n", but how do we know that such a thing even EXISTS?

4) At the end of the proof, they say "show x_{n}-> a", but HOW??

2. Relevant equations

N/A

3. The attempt at a solution

N/A

Can someone please explain the proof in greater detail?

Any help is much appreciated! :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: There exist a sequence x_n E S s.t. x_n -> sup S

**Physics Forums | Science Articles, Homework Help, Discussion**