Question on wikipedia's proof of rolles theorem 
Here is this proof. It basically says that if a function has a maximum, then this quotient, for $h>0$ is $$\frac{f(c+h)-f(c)}{h}\le0$$
That's ok to me, because if $c$ is the maximum point, then this quotient representes the 'almost tangent line' to the right. But what I can't understand, is why 
$$\frac{f(c+h)-f(c)}{h}\le0\implies \lim_{h\to 0^+}\frac{f(c+h)-f(c)}{h}\le0$$
It does not make sense to me. Then, above it, the article says the limit must be minus infinity. I cannot understand this. Why the $\lim$ is $-\infty$??? I can't even think about it, I'm so confused

$$
\Big(\frac{f(c+h)-f(c)}{h}\le0\text{ for all }h>0\Big)\implies \lim_{h\to 0^+}\frac{f(c+h)-f(c)}{h}\le0
$$
In other words, the limit of something never exceeding $0$ cannot exceed $0$.  Suppose the limit were $\ell>0$.  Let $\varepsilon=\ell/2$.  Then making $h$ close enough, but not equal, to $0$ and greater than $0$ would assure that the thing approaching the limit is within a distance $\varepsilon$ of the limit $\ell$.  But anything differing from $\ell>0$ by less than $\ell/2$ must be positive.
