Prove that every Cauchy sequence in $\mathbb{C}$ is bounded.
In $\mathbb{R}$, this is a sketch of the proof that I recall:
Let {${a_k}$} be Cauchy in $\mathbb{R}$, since $1\in\mathbb{R}$, $\exists N$ s.t. $\forall m,n>N$, $|a_n-A_N|<1\rightarrow$$|a_n|-|A_N|<|a_n-A_N|<1\iff|a_n|<1+|a_N|,\forall n>N-1$. Let $M = \max{|a_1|,|a_2|,\ldots,|a_N-1|,1+|a_N|}$. Then, $M$, $-M$ bound {$a_k$}.
A sequence is bounded in $\mathbb{C}$ if $\exists R\in\mathbb{R}$ and an integer $N$ s.t. $|z_n|<R$ $\forall, n>N$. Here's my attempt at the proof at hand then:
Let {${z_n}$} be Cauchy in $\mathbb{C}$. I want to show that there exists an R s.t. that definition above is satisfied. Is this R just the $M$ from the proof in $\mathbb{R}$?
Best Answer
To say that $-M$ and $M$ are respectively lower and upper bounds on the sequence $\{a_k\}$ is the same as saying $M$ is an upper bound on the sequence $\{|a_k|\}$. Think about how all that applies to $\mathbb{R}$ and then to $\mathbb{C}$.