Sum of Harmonic Numbers approaches Harmonic Number of Product of Indices

Theorem
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let:
 * $\map T {m, n} := H_m + H_n - H_{m n}$

where $H_n$ denotes the $n$th harmonic number.

Then as $m$ or $n$ increases, $\map T {m, n}$ never increases, and reaches its minimum when $m$ and $n$ approach infinity.

Proof
Similarly:
 * $\map T {m, n + 1} - \map T {m, n} \le 0$

From Approximate Size of Sum of Harmonic Series, the limiting value of $\map T {m, n}$ is the Euler-Mascheroni constant:


 * $\ds \lim_{m, n \mathop \to \infty} H_m + H_n - H_{m n} = \gamma$