Using Overpowered Theorems to Solve Easy Problems

I have known someone who liked to invoke Fubini's theorem to justify using $$ \sum_{m=a}^b \sum_{n=c}^d f(m,n) =\sum_{n=c}^d \sum_{m=a}^b f(m,n) $$

You can use forcing and then Shoenfield's theorem to prove the following theorem:

There exists a continuous function from $\Bbb R$ to $\Bbb R$ which is nowhere differentiable.

Essentially the idea is to define a countable forcing whose conditions are continuous approximations for our function, then the generic filter (i.e., our Cohen real) is easily continuous but nowhere differentiable. To get this as a proof, rather than a consistency proof, note that the statement "There is a continuous function which is nowhere differentiable" is a $\Sigma^1_2$-statement, so by absoluteness it was true in the ground model, which was arbitrary and therefore we have proved the wanted statement.

I found this one in a very old putnam mock test:

Show that the sum of two consecutive positive cubes is never a cube.