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We have a square grid and we want to go from point $A$ to $B$, we can only go up or right. There's the first path depicted in red below, now let's half each step, so we get the blue path, halving again we get the green path, then the yellow and so on (Excuse the crappy drawing below, I did it with paint, each 'new direction' is supposed to be half of the previous one).

My question is, as the halving 'algorithm' tend to infinity, it appears closer and closer to the diagonal line which is $a\sqrt{2}\ $ but the path's total length can't be anything else but $2a$.

What is wrong here ? Why is the limit false ?

Note: suppose the grid is over $\mathbb R^2$.

enter image description here

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  • $\begingroup$ Path length is not a continuous function. For any $\varepsilon>0$, you can find two paths $p_1$ and $p_2$ that are $\varepsilon$-close together ($|p_1(t)-p_2(t)|<\varepsilon$ for $t\in [0,1]$, say) but whose lengths differ by a nonvanishing amount. $\endgroup$ – mjqxxxx Mar 17 '14 at 21:24
  • $\begingroup$ @mjqxxxx what do you mean by non vanishing amount ? $\endgroup$ – GinKin Mar 17 '14 at 21:47
  • $\begingroup$ Bounded below by a nonzero constant (not dependent on $\varepsilon$). $\endgroup$ – mjqxxxx Mar 18 '14 at 20:28
  • $\begingroup$ Where are you getting "bounded below by a nonzero constant" approach/Ansatz? I'm guessing you're applying a general method from a greater, more general treatment of something. Could you point me to some textbook that deals with this? Basically, I'm not understanding the significance of a lower nonzero constant not dependent on ε. $\endgroup$ – 147pm Dec 14 '17 at 18:37
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Because that fact that $f_n(x)$ converges to $f(x)$ pointwise doesn't imply that every property shared by the $f_n$ carries over to $f$. There are many examples of properties that don' carry over to pointwise limits - in fact, most properties don't.

For example, all the $f_n$ could be continuous (they aren't in your case, but just imagine they were), yet $f$ could be discontinuous (yours isn't, but again, imagine it was). An example where this happens are the function $f_n(x) = x^n$ on the interval $[0,1]$. This is a sequence of continuous (even differentiable!) functions, yet its pointwise limit $$ f(x) = \begin{cases} 0 &\text{if $0 \leq x < 1$} \\ 1 &\text{if $x = 1$} \end{cases} $$ isn't continuous.

Or of course, as you observed, all the $f_n$ can have the same arc length, yet $f$ can have a different arc length.

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  • $\begingroup$ Same question as I asked above: Can you point me to a text that talks like this about such issues? There seems to be quite a bit of good math packed into this innocent-seeming question. $\endgroup$ – 147pm Dec 14 '17 at 18:40

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