Here are two proofs, firstly:

```
x = 0.999...
10x = 9.999...
    = 9 + 0.999...
    = 9 + x
9x = 9
x = 1

```

And secondly:

```
x = 1 - 1 + 1 - 1 + 1 - 1 ...
  = 1 - (1 - 1 + 1 - 1 + 1 ...
  = 1 - x
2x = 1
x = 1/2

```

The fourth line in the first and the third line in the second use the same trick. Does the legitimacy of the first proof guarantee that of the second?