I'm trying to understand Jensen's inequality, but I'm stuck understanding a lemma.
Let $\phi$ be a convex function, meaning $$\forall \lambda \in [0,1]:\lambda \phi(x) + (1-\lambda)\phi(y) \geq \phi(\lambda x + (1-\lambda)y)$$ show that:
$(i)$There is a linear function $l_a$ for any point $a$ s.t $l_a(x) \leq \phi(x)$ and $l_a(a) = f(a)$
$(ii)$If \phi is strictly convex then $l_a(x) = f(x) \iff x=a$
in Durret's book "Prbability: theory and examples" following solution is suggested:
By convexity we have $$\lim_{h \downarrow 0}\frac{\phi(a)-\phi(a-h)}{h} \leq \lim_{h \downarrow 0}\frac{\phi(a+h)-\phi(a)}{h}$$ and both limits exist since they are monotone. choosing $\beta$ between two limits and considering $l_a(x) = \beta(x-a)+a$ yields the desired conclusion.
In Durret's proof, even though I can see that both sequences are monotone, but existence of limit requires that they be bounded below(at least for RHS we need to show that it is bounded below otherwise they may both be equal to $-\infty$).
Also assuming existence of limits, I don't know how to conclude $(ii)$ in the lemma.