$H(p) \le H(q) + KL(p, q)$? Let $H(p) = \sum_i p_i\log\frac{1}{p_i}$ be the entropy of $p$
and $KL(p, q) = \sum_i p_i\log\frac{p_i}{q_i}$ be the KL divergence between $p$ and $q$. Does it hold that $H(p) \le H(q) + KL(p, q)$?
 A: This is false. The inequality can be written as 
$$
\sum_ip_i\log p_i-\sum_iq_i\log q_i+\sum_ip_i(\log p_i-\log q_i)\ge0\;.
$$
At $q_k=p_k$, the left-hand side is $0$. Differentiating it with respect to $q_k$ yields
$$
-1-\log q_k-\frac{p_k}{q_k}\;,
$$
and evaluating at $q_k=p_k$ yields
$$
-2-\log p_k\;.
$$
Due to the constraint $\sum_kq_k=1$, we can't vary the $q_k$ arbitrarily, but unless the derivatives are all equal (i.e. $p_k$ is constant), we can find a direction in which the left-hand side decreases. For instance, for $p_1=\frac23$, $p_2=\frac13$, it decreases with increasing $q_1$, and the inequality is violated e.g. for $q_1=\frac34$, $q_2=\frac14$:
$$
\frac23\log\frac23+\frac13\log\frac13-\frac34\log\frac34-\frac14\log\frac14+\frac23\left(\log\frac23-\log\frac34\right)+\frac13\left(\log\frac13-\log\frac14\right)
\\
=\log2\left(\frac23+\frac32+\frac12+\frac23+\frac43+\frac23\right)-\log3\left(\frac23+\frac13+\frac34+\frac23+\frac23+\frac13\right)
\\
=\frac{16}3\log2-\frac{41}{12}\log3\approx-0.06\lt0\;.
$$
