15. (25 points) Given that for any $x \geqslant 1$,
$$
\ln x - a\left(1 - \frac{1}{x}\right) \geqslant 0 \text{. }
$$

Find the range of real number $a$.
15. From the problem, we know that $x \ln x - a(x-1) \geqslant 0$.

Let $f(x) = x \ln x - a(x-1) \ (x \geqslant 1)$.
(1) When $a \leqslant 1$, since $x \geqslant 1$, we have
$$
f'(x) = \ln x + 1 - a \geqslant 0 + (1 - a) \geqslant 0.
$$

Thus, $f(x)$ is monotonically increasing on the interval $[1, +\infty)$, i.e., $f(x) \geqslant f(1) = 0$, which satisfies the problem's condition.
(2) When $a > 1$, if $1 < x < \mathrm{e}^{a-1}$, then
$$
f'(x) = \ln x + 1 - a < 0.
$$

Thus, $f(x)$ is monotonically decreasing on the interval $\left(1, \mathrm{e}^{a-1}\right)$, i.e., $f(x) < f(1) = 0$, which does not satisfy the problem's condition.
In summary, $a \leqslant 1$.