$(a_n)_n$ is Cauchy iff $(\Re(a_n))_n$ and $(\Im(a_n))_n$ are Cauchy

Question

Let $(a_n)_n$ be a Cauchy sequence in $\mathbb C$. I want to show

$(a_n)_n$ is Cauchy $\iff$ the real part $(\Re(a_n))_n$ and the imaginary part $(\Im(a_n))_n$ are both Cauchy

Answer 1

In fact, any finite collection of Cauchy sequences is also a Cauchy sequence and the reason is exactly what you said, but a little preciser. First you must define a metric on any singleton Cauchy sequence (since metrics are equivalent it doesn't matter which to choose) then combining them together to derive a general metric, i.e. assume a long-term sequence has been defined as $a_{n}=(a_{1,n},a_{2,n},...,a_{k,n})$ for some k. Define then a unique metric d(x,y) on the metric space in which $a_{i,n}$s are defined (for example R). Taking on general metric $d_{total}(x,y)=\sqrt {d(x_{1},y_{1})^{2}+d(x_{2},y_{2})^{2}+...+d(x_{k},y_{k})^{2}}$ (or whatever metric you want for example on $R^{k}$) and using triangle inequality the expected result could easily be concluded.