How to strictly mathematically prove that definition is wrong? I started to learn calculus by myself. First chapter of my textbook is about the limit of the sequence. I did all exercises in my textbook except one problem. There is some special problems in the end of this chapter: you need to find a mistake in the given definition. The last one is very weird. I don't understand how to strictly mathematically prove why this definition is wrong:
$L(a_n)$ - length of the curve $a_n$.
$D(a_n(P),S_{AB})$ - distance between point $P \in a_n$ and segment $S_{AB}$ (perpendicular from the point P to the segment $S_{AB}$). 
Definition: Sequence of smooth continuous curves  $a_n$ is called an approximation for segment $S_{AB}$ if:


*

*All curves $a_n$ begins at point A and ends at B. 

*For any $m<n, \{m,n\} \in \mathbb{N}, \ L(a_m) \geq L(a_n)$.

*For each $\epsilon >0$ there exists a natural number $N$ such that, for every $n\geq N$, for every points $P \in a_n$  we have $D(a_n(P),S_{AB})<\epsilon$.

*If (1-3) true then the sequence of smooth continuous curves $a_n$ is an approximation for a segment $S_{AB}$, their length tends to the limit L, which is length of a segment $S_{AB}$.

It is definitely wrong. With this definition we can prove that $5=4$. 
May be we should change in 2) that $L(a_m) > L(a_n)$? Or this is unfixable?  
 A: The definition is in principle fixable (after all, one could fix it by giving the standard definition), but there is a fundamental flaw: the approximating curve $a_n$ can be near the given curve in the sense of $(3)$, but can have length much greater than the length of the curve.  
The definition of length needs to give the "right answer" for simple cases where we have a good intuition, in particular for straight line segments. However, if we take the diagonal of a $1\times 1$ square, we can find a sequence of zig-zag straight line segments, with each straight line segment parallel to a side of the square, which is close to the diagonal in the sense of $(3)$, and always has total length $2$.  
True, these zig-zag paths have the property that $L(a_n)$ is constant for all $n$. But replacing $L(a_m)\ge L(a_n)$ for $m\le n$ by $L(a_m)\gt L(a_n)$ will not help much.  A small modification of the zig-zag paths can make their lengths strictly decreasing, with limit any number in the interval $[\sqrt{2},2)$.  
Note that not all of the "definition" is a definition. Assertion $(4)$ is in fact a theorem statement. That "theorem" happens to be false. The limit is very dependent on the sequence of "approximating curves" that we choose. 
