Consider the following theorem: The sum of two even integers is also an even integer.
What, if anything, is wrong with the following attempt at proving this theorem?

Proof.

Pick any two even integers, and write them as 2x and 2y. 2x + 2y can be rewritten as 2(x + y). Since (x + y) is the sum of two integers, which is also an integer, multiplying it by 2 must give an even integer. So 2(x + y) is even. Thus, the sum of two even integers is also even.

Question

Consider the following theorem: The sum of two even integers is also an even integer. 
 What, if anything, is wrong with the following attempt at proving this theorem? 
 
 Proof. 
 
  Pick any two even integers, and write them as 2x and 2y. 2x + 2y can be rewritten as 2(x + y). Since (x + y)  is the sum of two integers, which is also an integer, multiplying it by 2 must give an even integer. So 2(x + y) is even. Thus, the sum of two even integers is also even.

Answer

The proof is not verifiable - it only proves the result for two specific even integers, not all even integers.

Answer

The proof is not logical - it doesn't follow that the expression 2(x + y) is even.

Answer

The proof is not logical - it doesn't follow that the two integers selected are even.

Answer

There is nothing wrong with the proof.

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