Continuing in the same vein here are two one sentence proofs.
Theorem: For we have .
Proof: The coefficient of in the left hand side of the given expression, i.e. in , is obtained by selecting exactly parenthesis out of the total available to yield the , (the remaining parenthesis yield the ), which can be done in ways following which the coefficient is exactly .
Theorem: is irrational.
Proof: If is in lowest terms then is in lower terms.
Remark: The above proof generalizes to the case when is a non-square positive integer. Indeed, if then assuming that is in lowest terms also implies that is in lower terms.