Prove or Disprove If xn is a convergent sequence of real num

Prove or Disprove:

If x_n is a convergent sequence of real numbers, then x_n is a Cauchy sequence of real numbers.

Let M=A,d be a metric space

Every convergent sequence of real numbers is a cauchy seq of real numbers.

Let xn be a sequence in A that converges to the limit l?A.

Let ?>0.

Then also ?2>0.

Because xn? converges to l, we have:

?N:?n>N:d(xn,l)<?2

In the same way:

?m>N:d(xm,l)<?2

So if m>N and n>N, then:

This completes the proof that xn is a cauchy sequence.

d(xn,xm)

d(xn,l)+d(l,xm)

(by the Triangle Inequality)

?2+?2

(by choice of N)