Convincingly Wrong Mathematical Proofs & Arguments

Mistakes are disappointing, but sometimes they can be entertaining. In this post, I will give a list of incorrect mathematical arguments, that are not that easy to notice, especially if you are in a hurry. A few of these delicately wrong arguments were given by me at some points of my life. I will not point out which ones, though :)

1. An incorrect proof of the Baire category theorem for the set of real numbers

2. Does the central limit theorem contradict the law of the iterated logarithm?