Monotonicity of convex function

[hellbike]

$$f:(a,\infty)->R$$
i want to prove, that, if function is convex, then:

if exist $$x_1 \in R$$, exist $$x_2>x_1$$ : $$f(x_2)>f(x_1)$$
then:
for all $$x_3>x_2$$ for all$$x_4>x_3$$ : $$f(x_4)\ge f(x_3)\ge f(x_2)$$

in other words:
convex function is either decreasing on whole domain, or it starts to increase from some point and then is increasing from that point to the end of domain

We don't assume that function is differential.

With assumption that $$f(x_2)>f(x_1)$$ i should get something from definition of convex function, but i don't know how to do it.

I'm asking for some tip.

 

[g_edgar]

First show that convex implies this: If $a < b < c$, then
$$\frac{f(b)-f(a)}{b-a} \le \frac{f(c)-f(b)}{c-b}$$