# A sequence \({x_n}_{n=1}^{\infty}\) of points in a metric space \((E,d)\) is a Cauchy sequence if

A sequence \({x_n}_{n=1}^{\infty}\) of points in a metric space \((E,d)\) is a Cauchy sequence if

for every \(\epsilon>0\), there exists a positive integer \(N\) such that \(x_n\in B(x,\epsilon)\) for all \(n\in N\) where \(B(x,r)={y\in E:d(y,x)<\epsilon}\)

\(X_n_k+1>X_n_k\) and \(n_k\geq k, k=1,2,cdots\) and \(n_k:\mathbb{N}\rightarrow\mathbb{N}

and only if its component sequence converges

—>> for any \(\epsilon>0\), there exists an integer \(N_0>0\) such that for all \(m,n>N_0\) we get that \(d(x_n,x_m)<epsilon\)