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Let $p$ and $g$ be probability mass functions on a countable alphabet, with $g_i>0$ whenever $p_i>0$. If the cross-entropy $C=-{\sum }_i{p}_i{\log }_2{g}_i$ is finite, then $H(p)$ is finite, $H(p)\le C$, and $D(p\Vert g)=C-H(p)$. To avoid assuming the desired finiteness, first apply $\ln u\le u-1$ with $u=g_i/p_i$ to each positive $p_i$. For a finite initial segment,Zero coordinates satisfy this inequality too. Monotone convergence of the nonnegative information entropy summands and convergence of both total masses give $H(p)\le C<\infty$. Both logarithmic sums are then finite, justifying their difference and the relative entropy identity. This supplies a rigorous route to the maximum entropy distribution on the nonnegative integers without presupposing finite information entropy for every competing law.

For the maximum entropy distribution on the nonnegative integers, use the geometric distribution counting failures before the first success. Write $r=a/(a+1)$ and defineThe geometric series gives ${\sum }_i{g}_i=1$. Termwise differentiation of a power series applied to that geometric series for $0<r<1$ gives ${\sum }_{i}i{r}^i=r/(1-r{)}^2$, hence ${\sum }_{i}i{g}_i=r/(1-r)=a$.

For any competing probability mass function $p$ of mean $a$, its cross-entropy against $g$ isThis is finite. The fact that finite cross entropy bounds countable information entropy must be justified without assuming in advance that $H(p)$ is finite. Use $\ln u\le u-1$ with $u=g_i/p_i$ when $p_i>0$. Including zero terms by their limiting convention yields, for every finite $N$,Let $N\to \infty$. The information entropy partial sums increase, the cross-entropy partial sums converge to the finite value above, and the final mass difference tends to zero. This proves finiteness and the upper bound. We may now writeRelative entropy nonnegativity has equality exactly when $p=g$. Therefore the displayed geometric probability mass function is the unique maximizer, andThe mean constraint is essential: without it, uniform distributions on longer initial intervals have unbounded information entropy.

Write $p=(1+\mu {)}^{-1}$. Since $X$ and the geometric random variable $Z$ have the same expected value $\mu$,This is also the entropy deficit relative to the maximum entropy distribution on the nonnegative integers.

The random variables $X$ and $-Z$ are independent, so part a gives $H(X-Z)\ge H(-Z)=H(Z)$. Consequentlywhich is the required bound in terms of the Entropic Ruzsa distance.

For $\mu >0$, let $Q$ be the geometric distributionIf $P$ is the law of $X$, Gibbs inequality givesThereforeFor $\mu =0$, the nonnegative random variable $X$ is zero almost surely and both sides vanish. This proves that the maximum entropy distribution on the nonnegative integers with fixed expected value is geometric.

Write ${ϵ}_k=E_0(k+\frac{1}{2})$ and use Lagrange multipliers $\alpha ,\beta$ for the particle-number and energy constraints. The stationary equations areThus $n_k=e^{-\alpha }{e}^{-\beta {ϵ}_k}$. Finite normalization over infinitely many levels requires $\beta >0$. Put $q=e^{-\beta E_0}$, so $0<q<1$. The geometric series givesThereforeFor the energy constraint, use $n_k=N(1-q)q^k$ and $\sum k{q}^k=q/(1-q{)}^2$, obtained by differentiating the geometric series. This yieldsRecognizing the hyperbolic cotangent givesThe strict positivity of all occupations requires $E>N{E}_0/2$; the paper's positivity assumptions alone do not guarantee this feasibility condition. For admissible energy the multiplier is uniquely $\beta =E_0^{-1}\ln [(2E+N{E}_0)/(2E-N{E}_0)]>0$. Equality $E=N{E}_0/2$ is only the limiting state with all occupation in level zero, outside the stipulated strictly positive family; lower energy is impossible.

For rigor beyond formal multiplier equations, $p_k=n_k/N$ has fixed mean $E/(N{E}_0)-1/2$. The objective differs from $N$ times its information entropy only by the fixed constant $N-N\ln N$. The maximum entropy distribution on the nonnegative integers is uniquely geometric at that mean, confirming the stationary family and its maximum character.