Let $C_n$ be a sequence of rectifiable simple closed curves in $\mathbb{R}^2$ that converge to a rectifiable simple closed curve $D$ in the Hausdorff topology. It is easy to construct examples where
$$\limsup_{n \mapsto \infty} \text{length}(C_n) \neq \text{length}(D).$$
Question 0: I actually do not know any examples where the limit on the LHS does not exist. Can anyone provide one?
Question 1: In all the examples I know of, we have
$$\limsup_{n \mapsto \infty} \text{length}(C_n) \leq \text{length}(D).$$$$\limsup_{n \mapsto \infty} \text{length}(C_n) \geq \text{length}(D).$$
Must this always hold?
Question 2: If the $C_n$ bound convex regions, must we have
$$\lim_{n \mapsto \infty} \text{length}(C_n) = \text{length}(D)?$$
Question 3: Let $R_n$ be the region bounded by $C_n$ and let $U$ be the region bounded by $D$. Must we have
$$\lim_{n \mapsto \infty} \text{area}(R_n) = \text{area}(U)?$$
Edit: In the original version, I had the inequality in Question 1 backwards (thanks to Emil Jeřábek and kaleidoscop for pointing that out!). This has now been corrected. The examples I had in mind were a little easier than the ones in kaleidoscop's answer. Namely, let $C_n$ consist of the union of the sets
$$\{\text{$(t,0)$ $|$ $0 \leq t \leq 1$}\}$$
and
$$\{\text{$(1,t)$ $|$ $0 \leq t \leq 1$}\}$$
with a "staircase" that starts at $(0,0)$, goes $1/n$ up, then goes $1/n$ to the right, then $1/n$ up, then $1/n$ to the right, etc, ending at $(1,1)$. Each $C_n$ has length $4$. However, the $C_n$ converge to the union of the sets
$$\{\text{$(t,0)$ $|$ $0 \leq t \leq 1$}\}$$
and
$$\{\text{$(1,t)$ $|$ $0 \leq t \leq 1$}\}$$
and
$$\{\text{$(t,t)$ $|$ $0 \leq t \leq 1$}\},$$
which has length $2+\sqrt{2}$.
