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.