Is the Proof Valid for the Convergence of the Sequence x_n?

[bedi]

Homework Statement

Let x_n be a convergent sequence with a ≤ x_n for every n, where a is any number. Prove that a ≤ lim x_n when n→∞.

Homework Equations

Definition of limit. The usual ε, N stuff.

The Attempt at a Solution

Let lim x_n = x and choose ε=x_n-a. Hence we have |x_n - x| < x_n - a which shows that -x<-a and thus x>a.

Is this valid?

 

[micromass]

Homework Statement

Let x_n be a convergent sequence with a ≤ x_n for every n, where a is any number. Prove that a ≤ lim x_n when n→∞.

Homework Equations

Definition of limit. The usual ε, N stuff.

The Attempt at a Solution

Let lim x_n = x and choose ε=x_n-a.

You do realize that \varepsilon depends on n?? For a different n, you'll get a different \epsilon. Do you really want that??
Also, why is \varepsilon&gt;0? Specifically, why is it nonzero?

 

[tiny-tim]

hi bedi!

(try using the X₂ button just above the Reply box )

Let lim x_n = x and choose ε=x_n-a

but ε has to be independent of n

(ooh, micromass beat me to it! )

 

[bedi]

Yes, you are right. However I still can't see the solution :(

 

[tiny-tim]

try assuming the contrary

 

[bedi]

Alright, so this will imply that x_n converges to both a and x which is a contradiction. Am I right?

 

[micromass]

Alright, so this will imply that x_n converges to both a and x which is a contradiction. Am I right?

Take x_n=2 for all n and take a=0. Then certainly x_n does not converge to a. So no, you're not right.

 

[bedi]

But I assumed that a>x. So -a<-x and x_n-a<x_n-x<ε. Hence x_n-a<ε ?

 

[micromass]

But I assumed that a>x. So -a<-x and x_n-a<x_n-x<ε. Hence x_n-a<ε ?

Although the idea is there, the proof is still not very nice. For example, what is \varepsilon?? You got to say things like this, not just introduce them without telling anybody what it is.

 

[tiny-tim]

But I assumed that a>x …

… what is \varepsilon?? You got to say things like this, not just introduce them without telling anybody what it is.

bedi, you know this is supposed to be a delta,epsilon proof …

so you must define delta, and you must define epsilon​