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

$$\limsup_{n \mapsto \infty} \text{length}(C_n) \geq \text{length}(D).$$

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 $2+\sqrt{2}$.