Question

Show that if a sequence of complex numbers tends to zero, then the sequence of absolute values tends to zero too, and vice versa. Hint${\mathbf{a}}_{\mathbf{n}}\mathbf{+}{\mathbf{ib}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}$means $\mathbf{}\mathbf{}{\mathbf{a}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}\mathbf{}$and${\mathbf{b}}_{\mathbf{n}}\mathbf{\to }\mathbf{0}\mathbf{.}$

Short answer

It has been proved that${\mathrm{a}}_{\mathrm{n}}+{\mathrm{ib}}_{\mathrm{n}}\to 0$ means

Step-by-step solution

Given Information.

It has been given that the complex numbers tends to zero.

Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of $\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{}$in which x is the real part and y is the imaginary part.

Solve the series.

Assume a sequence of complex number.

$$\begin{gathered} S=\sum C_n \\ S=C_1+C_2+C_3+\cdots \end{gathered}$$

A complex number has a real and an imaginary part.

$$\begin{gathered} S=\sum C_n \\ S=\sum a_n+i{b}_n \\ S=\sum a_n+i\sum b_n \end{gathered}$$

If tends to be zero thentends to be zero and this is only possible when must be zero.