Whats wrong with this proof? (infinite sequences)

So I just watched a video where they explained that the sum of all natural numbers is $-1/12$. However, there was an interesting comment:

S1 = 1 + 2 + 3 + 4 + 5 ... = - 1/12
S1 - S1 = 
        1 + 2 + 3 + 4 + 5 + 6 ...
             - 1  - 2  - 3 -  4 -  5 ...
=      1 + 1 + 1 + 1 + 1 + 1 ...
Since S1 - S1 = - 1/12 - (- 1/12) = 0
It follows that 1 + 1 + 1 + 1 + 1 .... = 0
Let's name this sequence S2:

S2 = 1 + 1 + 1 + 1 + 1 ... = 0
Now let's subtract it from itself:
S2 - S2 = 
        1 + 1 + 1 + 1 + 1 ...
             - 1  - 1  - 1  - 1 ....
=      1
Given that S2 equals 0, we can also write this as:
0 - 0 = 1

For those who are wondering about the video: https://www.youtube.com/watch?v=w-I6XTVZXww It does the same trick of "shifting". With infinity and stuff things get tricky. What is wrong with this proof? I am a math student, so involved answers with limits is fine.