section_path: Complex analysis/Convergence
section_name: Convergence
paragraph_text: Convergence
  
  The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C {\displaystyle \mathbb {C} }, endowed with the metric d (z 1, z 2) = | z 1 − z 2 | {\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|} is a complete metric space, which notably includes the triangle inequality | z 1 + z 2 | ≤ | z 1 | + | z 2 | {\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|} for any two complex numbers z and z.