Let $ ABCD $ be a square with side length $ a $. Let $ s $ be a staircase from $ A $ to $ C $ with total length $ l $ and number of steps $ n $. It consists of perpendicularly alternating lines of length $ \frac{a}{n} $, as pictured here.
We see that $ l $ can be expressed as follows:
$$ l = \frac{a}{n} \cdot n + \frac{a}{n} \cdot n = \frac{a}{n} \cdot n \cdot 2 = 2a $$
and as such stays constant at $ 2a $.
Now let us imagine that the amount of steps is infinite, e.g. $ n = \infty $. Per definitionem, the staircase should now be the diagonal of the square with length $ l=a\sqrt{2} $ according to Pythagoras. This is paradoxical! According to the equation pictured above, it should have the length $ 2a $, not $ a\sqrt{2} $.
My question is:
Does $ l $ equal to $ 2a $ or $ a\sqrt{2} $?
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