Question

QQ

我想請問
原本fn只在所有compacts上一致收斂 D裡面會不會也包含沒辦法closed and bounded的部分
但我最後要證明的卻是在整個D上f都連續
這樣①③的選擇會不會只證明f在D中的任何compacts連續呢
如果會的話要怎麼改善證明
（高微證明小白求解..

Question 2 [10]
Let {fn (2)} be a sequence of continuous complex functions defined on D C C and f(z) be a complex
function defined on D. Prove by e-N definition that if fn(z) converges uniformly to f(z) on compacts
in D, then the limiting function f is also continuous.
0₁0. fn 3 f for z
For
every
Z
in K
E
> = N(6) € ₁ s.t. Unz N₁ | fn (2) - f(z)] < 1/10
N= .
in any compact sets K≤p.
@fn 75
20, 2 820 sit.lzx]<8>lfe)-f(x)]<-
continuons for z in
D.
For every z, & in D.
VE>O, we choose the above N and f
if Iz-*|< 8, then
s.t
| f(2)= f(a)| = |f₁z) - fn(z) + tn (2) - fn(x) + fn(x) – frasl
≤ | fn(z) - f1z)| + | fn(²2) - fn(x) |+| fricas) _flox) | =
D
E
1/4 + 1/3+1/²/3 =
3
=.E.

高微

Answer 1

Alstormeria5

Can you rephrase your question in English? So far I can tell is that complex plane is identical to R^2 space with the existence and similarity of metric.Therefore,compact subset of Complex number must be bounded and closed.