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QUESTION


Show that if $z_n=x_n+iy_n$  (for $n=1,2\dots)$, then $$\lim_{n
\rightarrow \infty} z_n=z$$ if and only if $$\lim_{n \rightarrow
\infty} x_n=x\ \ \ \ \ \hbox {and}\ \ \ \ \ \lim_{n \rightarrow
\infty} y_n=y.$$



ANSWER


Suppose that $lim_{n\rightarrow\infty} x_n=x$, and
$lim_{n\rightarrow\infty} y_n=y $. Then given $\epsilon>0,$ there
exists$n_0\in N$ such that $|x_n-x|<\epsilon/2,$
$|y_n-y|<\epsilon/2.$ If $x+iy=z$, and $x_n+iy_n=z_n$ then
$|z_n-z|\le|x_n-x|+|y_n-y|<\epsilon$. $lim_{n\rightarrow\infty}
z_n=z$. For the converse we just use $|Re(z)|\le |z|$ and
$|Im(z)|\le |z|$.




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