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hb123
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Homework Statement
Let {pn}n[itex]\in[/itex]P be a sequence such that pn is the decimal expansion of [itex]\sqrt{2}[/itex] truncated after the nth 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 {pn}n[itex]\in[/itex]P converges to p if ([itex]\forall\epsilon>0[/itex])([itex]\exists N \in P[/itex])([itex]\forall n\geq N[/itex])(| pn-p| < [itex]\epsilon[/itex]).
It is a Cauchy sequence if ([itex]\forall\epsilon>0[/itex])([itex]\exists N \in P[/itex])([itex]\forall n,m\geq N[/itex])(|pn-pm|< [itex]\epsilon[/itex]).
The Attempt at a Solution
I haven't gotten very far with this. Obviously, the sequence converges to p=[itex]\sqrt{2}[/itex]. Thus when working in the rationals, it doesn't converge and when working in the reals it does converge (since [itex]\sqrt{2}[/itex] 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.