Question

If z₁, z₂ ∈ C prove that

(i) |z₁ – z₂| ≤ |z₁| + |z₂|

(ii) |z₁ + z₂| ≥ |z₁| – |z₂|

(iii) |z₁ + z₂ + ….. + z_n| ≤ |z₁| + |z₂| + ……. + |z_n|

Answer 1

(iii) Now, we will prove (i) and (ii) by principle of mathematical induction.

Step I. Clearly the given standard is true for n = 1 and n = 2.

Step II. Let the given standard is true for n = k, i.e,

Hence, by the principle of mathematical induction the given statement is true for each positive integer n.