If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges

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Given a metric space [ilmath](X,d)[/ilmath]:


  • Let [ilmath](x_n)_{n=1}^\infty[/ilmath] be a Cauchy sequence with a convergent subsequence [ilmath](x_{k_n})_{n=1}^\infty\longrightarrow x\in X[/ilmath]
    • We wish to show that [ilmath](x_n)_{n=1}^\infty[/ilmath] converges (to [ilmath]x[/ilmath]), this means we must show:
      [ilmath]\forall\epsilon>0\ \exists N\in\mathbb{N}\ \forall n\in\mathbb{N}[n>N\implies d(x_n,x)<\epsilon][/ilmath]
      Note that by the triangle inequality of [ilmath]d[/ilmath] you should already be thinking [ilmath]d(x_n,x)\le d(x_n,x_\text{other})+d(x_\text{other},x)[/ilmath]
    • Let [ilmath]\epsilon>0[/ilmath] be given.
      • As [ilmath](x_n)_{n=1}^\infty[/ilmath] is Cauchy:
        • [ilmath]\exists N_1\in\mathbb{N}\forall n_1,m_1[n_1,m_1>N\implies d(x_{m_1},x_{n_1})<\frac{\epsilon}{2}][/ilmath]
          (As it is Cauchy we know that for whatever number [ilmath]>0[/ilmath] we like there exists an $N$, here we pick the number [ilmath]\frac{\epsilon}{2} [/ilmath])
      • As [ilmath](x_{k_n})_{n=1}^\infty[/ilmath] converges to [ilmath]x\in X[/ilmath]:
        • [ilmath]\exists N_2\in\mathbb{N}\forall n_2\in\mathbb{N}[n_2>N_2\implies d(x_{k_{n_2} },x)<\frac{\epsilon}{2}][/ilmath]
      • Choose [ilmath]N:=\text{Max}(N_1,k_{N_2})[/ilmath] (so the statements [ilmath]d(x_{k_{n_2} },x)<\epsilon[/ilmath] and [ilmath]d(x_{m_1},x_{n_1})<\epsilon[/ilmath] hold)
        • Let [ilmath]n\in\mathbb{N} [/ilmath] with [ilmath]n>N[/ilmath] be given
          • [ilmath]d(x_n,x)=d(x,x_n)\le d(x,x_{k_n})+\underbrace{d(x_{k_n},x_n)}_{\text{as }n\le k_n}<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon[/ilmath]
            Note: Here we use that for a subsequence [ilmath]n\le k_n[/ilmath]
      • This completes the proof


TODO: Create and link to the theorem that [ilmath]n\le k_n[/ilmath] for a subsequence.


  1. Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene