TITLE: What are some examples of subtle logical pitfalls?
QUESTION [6 upvotes]: Here's an example: 
Demonstrating that the assumption $A=B$ leads to a true statement is a vacuous truth. In order the show that $A=B$, prove that the difference $\Delta =A-B$ is zero.  The subtle change being that $\Delta$ is not assumed to be zero. 

What are some other examples of subtle logical pitfalls that the amateur Mathematician should be aware of?

Here is an specific argument that shows how assuming $A=B$ leads to absurdity.
$$
\begin{eqnarray}
2&=&1\\ 
2-1&=&1-1\\ 
1&=&0\\ 
\end{eqnarray} 
$$
$$
\begin{eqnarray}
a+b&=&a+b\\ 
a+1*b&=&1*a+b\\ 
a+0*b&=&0*a+b\\ 
a&=&b
\end{eqnarray} 
$$
A falsity implies anything. Assuming that the false statement is true implies that the two undefined objects $a$ and $b$ are equal, absurd. However, if we define the difference as $\Delta$, then a true statement is forced.
$$
\begin{eqnarray}
2-1&=&\Delta\\ 
\Delta &=&1\\ 
2&=&\Delta +1\\ 
2&=&2
\end{eqnarray} 
$$

REPLY [7 votes]: I think this is probably the subtlest point I have yet to encounter in set theory.
Let's assume the universe of sets satisfies $\sf ZFC$.
Fact I: For every $T_0$, a finite subset of $\sf ZFC$, we can prove in $\sf ZFC$ that $T_0$ has a model.
Fact II: The compactness theorem is provable from $\sf ZFC$, so if a theory $T$ is such that every finite fragment has a model, then $T$ has a model.
Fact III: If $\sf ZFC$ is consistent, then $\sf ZFC$ cannot prove its own consistency. In particular, if we assume that the universe of sets satisfies $\sf ZFC$, then it is impossible to prove that there is a set which is a model of $\sf ZFC$.
On first sight it seems that the first two facts should contradict the third! But the truth is that the first fact is proved in the meta-theory. And we cannot apply compactness arguments in the meta-theory, and have the results in the theory itself!
This pitfall is $\large\sf\text{confusing between meta-theory and theory}$. It is very easy to fall for it when making your first steps in set theory and consistency results.