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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:

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

  2. For any $m.

  3. 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 $D(a_n(P),S_{AB})<\epsilon$.

  4. 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?

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    @Thomas "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"2012-08-27

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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.

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    @Mike: For any *particular* sequence $(a_n)$ of approximating curves, the limit does exist (non-increasing sequence, bounded below). However, that limit is very dependent on the sequence $(a_n)$ chosen, so cannot serve as the definition of length: a definition should, at least in nice cases, give a definite answer. If $L$ is the actual length, by appropriate choice of $(a_n)$ satisfying $(1,2,3)$ we can get any limit in $[L,\infty)$.2012-08-27