You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into as many pieces as students come for lunch, including yourself.

The problem is, the knife is operated by the rotating door through which the students enter, so every time a student comes in, the knife comes down and you place the cut. There is no way for you to know if more students will come or not, so after each cut, the sausage should be cut into pieces of approximately equal length. 

So here the question - is it possible to place the cuts in a manner to ensure the ratio of the largest and the smallest piece is always below 2?

And if so, what is the smallest possible ratio?

Example 1 (unit is cm):

* 1st cut: 50 : 50 ratio: 1
* 2nd cut: 50 : 25 : 25 ratio: 2 - bad

Example 2

* 1st cut: 40 : 60 ratio: 1.5
* 2nd cut: 40 : 30 : 30 ratio: 1.33
* 3rd cut: 20 : 20 : 30 : 30 ratio: 1.5
* 4th cut: 20 : 20 : 30 : 15 : 15 ratio: 2 - bad

Sorry for the awful analogy, I think this is a math problem but I have no real idea how to formulate this in a proper mathematical way.