Let $C$ be a smooth curve in $\Bbb C$ and suppose that the sequence of continuous functions $f_n$ converges to $f$ uniformly on the curve $C$.
Show that $\int _C f_n(z) dz $ converges to $\int_C f(z) dz$.
Suppose that $f$ is a continuous function on a domain $\Omega $ in $\Bbb C$. Suppose that the continuously differentiable curves $C_n$ converge uniformly to the continuously differentiable curve $C$ in $\Omega$.
Show that $\lim\limits_{n\to \infty} \int _{C_n} f(z) dz\neq \int_C f(z) dz$ for some $C_n$ and $f$ continuous.
Now the first question I solved it like this:
Given $\epsilon >0$ there exists $m\in \Bbb N$ such that
$|f_n(z)-f(z)|<\epsilon $ forall $n\ge m$ and $\forall z\in C$.
Then $\left|\int _C f_n(z) dz-\int_C f(z) dz \right|\le \int _C \left|f_n(z)-f(z)\right|dz $
$< \epsilon \times $ length of the curve $C$ $\forall n\ge m$ and $\forall z\in \Bbb C$.
Is it correct?
I am stuck on the 2nd part and unable to make any progress. How should I find $C_n$?I need some help.