proof for Kullback Leibler divergence is nonnegative

Background

Kullback Leibler:
$$
KL(p||q) = \sum\nolimits_{k=1}^K{p_k}\ln\frac{p_k}{q_k} \geq 0
$$
is used to calculate the distance between two distributions where:

$$
\sum\nolimits_{k=1}^K{q_k}=1
$$

$$
\sum\nolimits_{k=1}^K{p_k} = 1
$$

$$
{p_k} \geq 0
$$

$$
{q_k} \geq 0
$$

Proof

$$
{p_k}·\ln\frac{p_k}{q_k} = -{p_k}·\ln\frac{q_k}{p_k}
$$

According to:

$$
-\ln x \geq {1 - x}
$$

we can get the following equation:

$$
-p_k\ln\frac{q_k}{p_k} \geq p_k(1 - \frac{q_k}{p_k})
$$

And go ahead:
$$
\sum\nolimits_{k=1}^K{p_k}\ln\frac{p_k}{q_k} \geq \sum\nolimits_{k=1}^K{p_k}-\sum\nolimits_{k=1}^K{q_k}\frac{p_k}{p_k} = 0
$$