The Diagonal Paradox

Shown here is a square with edges of length 1. The two orange edges will sum up to 2. Additionally, the length of the diagonal can be calculated using the Pythagorean Theorem. You know, A² + B² = C². That one. This gives √2 ≈ 1.41 for the diagonal.

Start carving out a stairway as shown in the pictures below. Realise that the total length of those orange lines will remain 2 no matter how many steps you chose to make!

From left to right, there are more (smaller) steps each time. But hang on a second: that orange line is quickly starting to look like the diagonal. Would it also approach 1.41 in length? If you made infinitely many steps, would its length turn out to be exactly √2?
No, not so, on both accounts.