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Consider an isosceles right-angled triangle as shown in the figure (top left). The length of its hypotenuse is $c$. The figure distinguishes both legs of the triangle, however, from now on let's assume, since it's an isosceles right-angled triangle, that $b=a$. Now, let's build a stair on the hypotenuse with steps of height $\frac{a}{n}$, and width $\frac{a}{n}$. If $n=1$ we have the picture of the figure (top centre), in which $d=a$ and $e=a$. Here the length of the stair is $d+e=2a$.

Notation: I will refer to each step in the stair as $s_k$ for some $k\in\mathbb N:0<k\leq n$.

If we continue doing the same procedure, we have that for some $n\in\mathbb N$ that the lenght $\ell(n)$ of the stair is:

$$\ell(n)=\sum\limits_{k=1}^n \text{lenght}(s_k) = \sum\limits_{k=1}^n2\frac{a}{n}=2a.$$

If we build a stair with infinitely small steps, why don't we end up with a straight line? Because if we did, we would be saying that $c=2a$, and by the pythagorean theorem we know that $c=\sqrt{2}a$. I appreciate your thoughts.

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    $\begingroup$ This is the same idea as in: math.stackexchange.com/q/12906/73324 $\endgroup$
    – vadim123
    Nov 17 '14 at 3:16
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    $\begingroup$ Agreed, it's the same idea. I'd also put out there that the distinction between $c = 2a$ and $c = a\sqrt{2}$ is analogous to the difference between taxicab distance and euclidean distance. If you haven't looked into taxi-cab geometry, it's relevant here. $\endgroup$
    – Nick H
    Nov 17 '14 at 3:25
  • $\begingroup$ Pretty cool! Didn't know about that. $\endgroup$ Nov 17 '14 at 3:30
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    $\begingroup$ @NickH: I believe your answer is most complete (taxicab vs Euclidean). Could you please copy it into an "answer" so that OP can accept it or we can upvote? $\endgroup$
    – Deepak
    Nov 17 '14 at 3:37
  • $\begingroup$ "why don't we end up with a straight line". In Euclidean Geo, a (straight) line is the shortest distance between 2 points. The path that passes by a and b has a total length of 2a. The hypotenuse, otoh, has length of sqrt(2)*a, known from Pythagoras Theorem (we can prove from basic principles). Thus the longer 2*a distance won't meet the definition of a (straight) line because it is provably not the shortest distance here. $\endgroup$
    – Jose_X
    Oct 30 '16 at 13:06

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