TITLE: "It looks straightforward, but actually it isn't"
QUESTION [58 upvotes]: In a previous topic, I asked about proof of statements which are simple but incorrect. 
Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I expect the answers to contain:

the statement;
why it looks easy to prove;
why actually it isn't.

REPLY [19 votes]: Theorem: 1 + 1 = 2  
Proof:  
Many pages of...  

Whitehead, Alfred North, and Bertrand Russell (1910, 1912, 1913).
Principia Mathematica, 3 vols, Cambridge: Cambridge University Press.
  Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3).  

Note that "The above proposition is occasionally useful."