In an answer some time ago I referenced a short proof that the constant $e$ is irrational by A. R. G. MacDivitt and Yukio Yanagisawa (The Mathematical Gazette , Volume 71 , Issue 457 , October 1987 , pp. 217)
What is the shortest proof that there is an irrational number?
I think the technique can be applied to prove that the constant $U_1=\sum_{k=1}^\infty \frac{1}{k.k!}$ is irrational.
Step 1: Suppose $U_1$ is rational; $U_1$ being defined by the infinite sum, $\sum_{k=1}^\infty \frac{1}{k.k!}$; that is:
$$\sum_{k=1}^\infty \frac{1}{k.k!}=\frac{m}{n}\tag{1},$$
where $m$ and $n$ are positive integers.
Step 2: Multiply both sides of (1) by $D(H_n).n!$, where $D(H_n)$ is the denominator of the Harmonic Number $n$ (simplified form for neatness, although this choice is not essential to the proof) thus
$$D(H_n).n!\left(1+\frac{1}{1!}+\frac{1}{2\times 2!}+\frac{1}{3\times 3!}+...+\frac{1}{n.n!} \right) \\+\frac{D(H_n)}{(n+1)^2}+\frac{D(H_n)}{(n+1)(n+2)^2}+\frac{D(H_n)}{(n+1)(n+2)(n+3)^2}+...=D(H_n)n!\,\frac{m}{n}\tag{2}$$
so that $S=\frac{D(H_n)}{(n+1)^2}+\frac{D(H_n)}{(n+1)(n+2)^2}+...$ is a positive integer, with $nS \ge \frac{D(H_n)}{(n+1)}$.
Step 3: Construct contradiction.
$nS\ge \frac{D(H_n)}{(n+1)}$ can be written $(n+1)S-\frac{D(H_n)}{(n+1)} \ge S$, implying that
$$\frac{D(H_n)}{(n+2)^2}+\frac{D(H_n)}{(n+2)(n+3)^2}+...\ge \frac{D(H_n)}{(n+1)^2}+\frac{D(H_n)}{(n+1)(n+2)^2}+...;$$
which after comparing both sides of the inequality term by term can immediately seen to be a contradiction. (Assuming I haven't made a mistake)
Assuming this is correct I am not sure if the same method can apply to the alternating sum $$U_2=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k.k!}$$
As the sum $U_2$ is unconditionally convergent, adjacent positive and negative terms in the sum can be brought together giving positive terms only, thus
$$U_2=\sum_{k=1}^\infty \frac{(2k-1)^2+2k}{(2k-1)2k(2k)!}$$
but unfortunately this gives a numerator depending on $k$ and I think this stops the same proof from working.
For added background
$U_1$ and $U_2$ are related to the Euler-Mascheroni Constant, $\gamma$, thus
$$U_1=-\gamma+Ei(1)$$ $$U_2=\gamma-Ei(-1)$$
where $Ei(x)$ is exponential integral.
The Euler-Gompertz constant $\delta$ is $$\delta=-e\,Ei(-1)=eU_2-e\gamma$$
Additional Comment
It should also be noted that
$$U_1=\sum_{k=1}^\infty \frac{1}{(2k-1)(2k-1)!}+\sum_{k=1}^\infty \frac{1}{(2k)(2k)!}$$
and
$$U_2=\sum_{k=1}^\infty \frac{1}{(2k-1)(2k-1)!}-\sum_{k=1}^\infty \frac{1}{(2k)(2k)!}$$
Therefore $U_2$ the number we are trying to prove irrational is equal to
$$U_2=U_1 -2\sum_{k=1}^\infty \frac{1}{(2k)(2k)!}$$
which shows how infuriatingly elusive irrationality proofs can be.
