Let X d be a metric space Show that every Cauchy sequence xn

Let (X, d) be a metric space. Show that every Cauchy sequence {x_n} X is bounded.

Solution

we need to prove that every cauchy sequence {x_n} in metrix space {X,d} is bounded.

let assume that x_n be a Cauchy sequence of (X, d)

hence we can say that by definition of cauchy sequence = 1

so we can say that there exists N such that,

d(x_m , x_n) < 1 for all m, n N

hence we can say that x_n B(x_N , 1) for all n N

now r = 1 + max{1, d(x₁, x_N), d(x₂, x_N), . . . , d(x_N-1, x_N)}

we can see that,

x_n B(x_N ; r) for all n

so we can say that x_n is bounded.