Determining if a sequence is convergent and/or a Cauchy sequence

[hb123]

Homework Statement

Let {p_n}_n\inP be a sequence such that p_n is the decimal expansion of \sqrt{2} truncated after the n^th decimal place.
a) When we're working in the rationals is the sequence convergent and is it a Cauchy sequence?
b) When we're working in the reals is the sequence convergent and is it a Cauchy sequence?

Homework Equations

A sequence {p_n}_n\inP converges to p if (\forall\epsilon&gt;0)(\exists N \in P)(\forall n\geq N)(| p_n-p| < \epsilon).
It is a Cauchy sequence if (\forall\epsilon&gt;0)(\exists N \in P)(\forall n,m\geq N)(|p_n-p_m|< \epsilon).

The Attempt at a Solution

I haven't gotten very far with this. Obviously, the sequence converges to p=\sqrt{2}. Thus when working in the rationals, it doesn't converge and when working in the reals it does converge (since \sqrt{2} is a real number but not rational). However, I'm stuck trying to prove this using the mentioned definitions, and can't get anywhere with trying to prove if the sequence is Cauchy.

 

[Dick]

Homework Statement

Let {p_n}_n\inP be a sequence such that p_n is the decimal expansion of \sqrt{2} truncated after the n^th decimal place.
a) When we're working in the rationals is the sequence convergent and is it a Cauchy sequence?
b) When we're working in the reals is the sequence convergent and is it a Cauchy sequence?

Homework Equations

A sequence {p_n}_n\inP converges to p if (\forall\epsilon&gt;0)(\exists N \in P)(\forall n\geq N)(| p_n-p| < \epsilon).
It is a Cauchy sequence if (\forall\epsilon&gt;0)(\exists N \in P)(\forall n,m\geq N)(|p_n-p_m|< \epsilon).

The Attempt at a Solution

I haven't gotten very far with this. Obviously, the sequence converges to p=\sqrt{2}. Thus when working in the rationals, it doesn't converge and when working in the reals it does converge (since \sqrt{2} is a real number but not rational). However, I'm stuck trying to prove this using the mentioned definitions, and can't get anywhere with trying to prove if the sequence is Cauchy.

Pick a specific ε, say ε=1/10000. How large does N have to be?

 

[hb123]

The thing is, though, I need to show that this is true for all ε, not just an arbitrary one. On the other hand, if I'm trying to prove that the series doesn't converge to a p, then I'd need to show that !(∀ϵ>0)(∃N∈P)(∀n≥N)(| pn-p| < ϵ), or (∃ϵ>0)(∀N∈P)(∃≥N)(|p_n-p| \geq ϵ). Just to clarify, P is the set of all positive integers.

 

[Dick]

The thing is, though, I need to show that this is true for all ε, not just an arbitrary one. On the other hand, if I'm trying to prove that the series doesn't converge to a p, then I'd need to show that !(∀ϵ>0)(∃N∈P)(∀n≥N)(| pn-p| < ϵ), or (∃ϵ>0)(∀N∈P)(∃≥N)(|p_n-p| \geq ϵ). Just to clarify, P is the set of all positive integers.

I know. I'm suggesting you think about the N corresponding to specific case ε=0.0001 to get you started. Working it out is really not much different from a general value of ε. And I'm talking about the Cauchy part of the proof here.