Proof: (related to "Convergent Real Sequences Are Cauchy Sequences") 

• By hypothesis, a given real sequence $(a_n)_{n\in\mathbb N}$ is convergent to the limit $a\in\mathbb R.$
• This means that for a given $\epsilon > 0$ there is an index $N\in\mathbb N$ such that $$|a_n-a| < \frac{\epsilon}2$$ for all $n\ge N.$
• Thus, by the triangle inequality $$\begin{align}|a_n- a_m|&=|a_n-a+a-a_m|\nonumber\\
  &\le |a_n-a|+|a-a_m|\nonumber\\
  &\le \frac{\epsilon}{2}+\frac{\epsilon}{2}\nonumber\\
  &=\epsilon\nonumber\end{align}$$ for all $n,m\ge N.$
• It follows that $(a_n)_{n\in\mathbb N}$ is a real Cauchy sequence.

| | | | created: 2020-07-11 16:01:49 | modified: 2020-07-11 16:02:10 | by: | references: [581]