Jonathan Sondow elegantly proves the irrationality of e in his aptly titled A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality (The American Mathematical Monthly, Vol. 113, No. 7 (Aug. - Sep., 2006), pp. 637, http://www.jstor.org/stable/27642006).

In his argument, he constructs a sequence of nested intervals $I_n$ for every $n \geq 1$, each of the form $[k/n!, (k+1)/n!]$, such that $\bigcap I_n = ${$e$}, with $e$ lying strictly between the endpoints of each $I_n.$ From this, we conclude that $e$ cannot be written as a fraction with denominator $n!$ for any $n \geq 1.$

Fact: Every rational number $p/q$ can be written as a fraction with a factorial in its denominator: $p/q = p(q-1)!/q!$.

Thus, we conclude that $e$ is irrational.

The reason this proof technique works so well with $e$ is, of course, related to the Maclaurin series for the exponential function, $e^x.$

That any rational number can be written in lowest terms is employed in other irrationality proofs (e.g., the classic proof for that of $\sqrt{2}$) but I had not seen the above fact drawn upon before reading this particular paper.

My question is: are there other examples of real numbers (which are not related to $e$ in some trivial way) whose irrationality can be proved using the Fact above?

Jonathan Sondow elegantly proves the irrationality of e in his aptly titled A Geometric Proof that e Is Irrational and a New Measure of Its Irrationality (The American Mathematical Monthly, Vol. 113, No. 7 (Aug. - Sep., 2006), pp. 637, http://www.jstor.org/stable/27642006).

In his argument, he constructs a sequence of nested intervals $I_n$ for every $n \geq 1$, each of the form $[k/n!, (k+1)/n!]$, such that $\bigcap I_n = \{e\}$, with $e$ lying strictly between the endpoints of each $I_n.$ From this, we conclude that $e$ cannot be written as a fraction with denominator $n!$ for any $n \geq 1.$

Fact: Every rational number $p/q$ can be written as a fraction with a factorial in its denominator: $p/q = p(q-1)!/q!$.

Thus, we conclude that $e$ is irrational.

The reason this proof technique works so well with $e$ is, of course, related to the Maclaurin series for the exponential function, $e^x.$

That any rational number can be written in lowest terms is employed in other irrationality proofs (e.g., the classic proof for that of $\sqrt{2}$) but I had not seen the above fact drawn upon before reading this particular paper.

My question is: are there other examples of real numbers (which are not related to $e$ in some trivial way) whose irrationality can be proved using the Fact above?