Will assuming the existence of a solution ever lead to a contradiction?

Solution 1:

Just the first thing that came to my mind... assume $A=\sum_{n=0}^{\infty}2^n $ exists, it is very easy to find $A $: note $A=1+2\sum_{n=0}^{\infty}2^n =1+2A $, so $A=-1$.

Of course, this is all wrong precisely because $A $ does not exist.

Solution 2:

Here is a "joke" due to Perron showing that assuming the existence of a solution is not always a very good idea:

Theorem. $1$ is the largest positive integer.

Proof. For any integer that is not $1$, there is a method to obtain a larger number (namely, taking the square). Therefore $1$ is the largest integer. $\square$

A good source is V. Blåsjö, The isoperimetric problem, Amer. Math. Monthly 112 (2005), 526-566.