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 an, and width an. 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 sk for some $k\in\mathbb N:0
If we continue doing the same procedure, we have that for some n∈N that the lenght ℓ(n) of the stair is:
ℓ(n)=n∑k=1lenght(sk)=n∑k=12an=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=√2a. I appreciate your thoughts.
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