Convergence Proof: Showing (\sqrt{x_{n}})\rightarrow0

[kathrynag]

Homework Statement

Let $$x_{n\geq}$$0 for all n in the natural numbers.
If ($$x_{n}$$)$$\rightarrow$$0, show that ($$\sqrt{x_{n}}$$)$$\rightarrow$$0.

Homework Equations

The Attempt at a Solution

So far, I have started with $$\left|\sqrt{x_{n}}-0\right|$$. Not sure if that's the right way to start.

 

[Office_Shredder]

You should probably start with the definition of convergence

 

[kathrynag]

A sequence converges to a real number a if for every positive $$\epsilon$$, there exists an N element of the natural numbers such that whenever n$$\geq$$N, it follows that $$\left|a_{n}-a\right|$$<$$\epsilon$$.

 

[╔(σ_σ)╝]

$$a^{2} \leq b^{2}$$ iff $$a \leq b$$

$$a,b \geq 0$$

Can you use this ?

 

[kathrynag]

Then the sequence is less than 0 and thus converges to 0

 

[╔(σ_σ)╝]

How did you arrive at such a conclusion? Btw what you said not correct.

How is the sequence less than zero ? In your definition $$x_{n} \geq 0$$.

 

[kathrynag]

so the sequence is greater than 0 because x is greater than 0.

 

[╔(σ_σ)╝]

All I wanted you to do wanted you to do was take the square root of both sides of the inequality...
$$x_{n} < \epsilon$$.

 

[Inferior89]

If g(x) --> A when x --> a then p(g(x)) --> p(A) when x --> a.

 

[╔(σ_σ)╝]

Hmm... what if p was the square root function and A was negative. ?

 

[Inferior89]

Well A needs to be in the domain of p for it to make sense. Guess I should have written that..

 

[╔(σ_σ)╝]

It's fine. Btw this theorem is not one of the 4 limit theorems given in most analysis books so I doubt the OP can use it. OP would need to prove it to use it.

 

[Inferior89]

We had 5 limit theorems when I did analysis in first year at uni. The proof is like 3 lines and not harder than the rest so I think it is strange it isn't standard at other places

 

[╔(σ_σ)╝]

I think you are referring to limits of functions not limit theorems. The limit theorems are for sequences and they are later generalized to functions.
I am taking analysis right now and the thoerem you mentioned is in the limit of functions section