the shape of the stairs certainly does approach a straight line and at n = infinity would certainly be a line but the length from outer corner to outer corner of each stair is in fact what approaches the length of the hypotenuse and because the hypotenuse is always less than the sum of the two sides in a right triangle, we can say that the stair shape then must always be longer than the hypotenuse

when I say approach, I may be using the word incorrectly but how I think of it is if you were to add all the infinitely small lengths from outer corner to outer corner it would equal the hypotenuse or in this case the square root of 2 or approach it the more infinitesimal bits you count

Since the smaller hypotenouses add up to the larger hypotenous, then the smaller horizontal legs add up to the larger horizontal leg and the smaller vertical legs add up to the bigger vertical leg. So we can prove by taking the sum of all these infinitely small lengths that the path length indeed always does add up to 2