Theorem: convergence of complex sequence

[sara_87]

Homework Statement

show that a sequence z_n (of complex numbers) converges if and only if it's real and imaginary parts converge.

Homework Equations

The Attempt at a Solution

if the real parts converge and the imaginary parts converge then using linearity, we have:
lim(real + imaginary)= lim(real) + lim(imaginary)

I think this is a wrong proof (and surely incomplete). does anyone have any ideas?

 

[tiny-tim]

Hi sara_87!

Just use an ordinary δ,ε proof, with δ = δ₁ + iδ₂ and ε = ε₁ + iε₂, and remember |a + ib| = √(a² + b²)

 

[sara_87]

hi tinytim,
thanks ,
ok so here it goes:
we have 'if and only if' so first I am going to 'try' and prove it in the first direction:
let z_n be a convergent sequence (and converges to z_c), then:
for all $$\epsilon>0$$ there exists $$\delta$$ such that $$\left|z_n-z_c\right|<\epsilon$$ for all n>$$\delta$$.
let z_c=a+ib:
$$\left|z_n-(a+ib)\right|=\left|z_n+(-a-ib)\right|\leq\left|z_n\right|+\left|(-a-ib)\right|=sqrt(a^2+b^2)$$
(i used the triangle inequality)
but now i don't know what to do.

 

[statdad]

Think about this:

If $$z_1 =e + f i$$ and $$z_2 = g + h i$$ are complex numbers, then

$$|e - g | \le |z_1 - z_2|$$

and

$$|f - h| \le |z_1 - z_2|$$

 

[sara_87]

i agree, but why would this help here, do i need to use this to put
$$\left|z_n\right|+\sqrt{a^2+b^2}$$ in terms of something that could be my $$\epsilon$$

 

[statdad]

Suppose you want to show that

$$z_n = x_n + y_ni \to z = x+yi$$
if, and only if

$$x_n \to x \text{ and } y_n \to y$$

If you know the complex sequence converges then the inequalities will show the two real sequences converge. If the real sequences converge, you can show the complex one does as well.

 

[sara_87]

oh right i see,
so first, to prove in the first direction:
let z_n be a convergent sequence, then
lim(z_n)=lim(x_n)+i*lim(y_n)

z_n-z_c=(x_n-x)+ i (y_n-y)
where z_c=(x,y) is the limit of the sequence z_n.
so:
$$\left|x_n-x\right|\leq\left|z_n-z_c\right|$$ and $$\left|y_n-y\right|\leq\left|z_n-z_c\right|$$
is this right? buti don't fully understand why these ineqaulities are true?