Componentwise convergence of complex sequences
A complex sequence converges to if and only if its real and imaginary parts converge: exactly when and .
Lemma 1.7
Let be a complex sequence. Then if and only if and .
Same criterion for Cauchy sequences
Proof
For , . If then and , so both parts converge to the corresponding part of .
Conversely, , so fixing and taking with and for respectively gives for all . Since is arbitrary and can be replaced by , the sequence converges.
The Cauchy statement is proved by the same two inequalities applied to in place of .
Related
Stated in
- Lemma 1.7§1.1 Basics
- Lemma 1.20§1.3 Cauchy Sequences