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
. 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?