Question

prove that every convergent sequence in R is a cauchy sequence

Answer 1

A sequence x1, x2, x3, ........ of real numbers is known as cauchy sequence, if for every positive real number ε > 0 , there is a positive integer N such that for all natural numbers m,n >N
$|x_m-x_n| < \epsilon$

now, we have to prove that : Every convergence sequence in R is a cauchy sequence .

proof : Let {an} is the sequence in R , that converge to limit l ∈ R,
Let ε > 0 { every positive real number }
then, also you can say ε/2 > 0

we know, if {an} converges to l . then, we have
∃N:for all n > N : d(an, l) < ε/2 --------(1)
so, if n>N and m > N then,
d(an, l) + d(l,am) ≥ d(an,am) { by triangle inequality }

from equation (1)
ε/2 + ε/2 ≥ d(an,am)
ε ≥ d(an,am) ,
by choice N , ε > d(an,am)
from the definition of cauchy , it is clear that
{an} is a cauchy sequence.

hence it is clear that every convergence in R is a cauchy sequence .