Rising Sun Lemma Spivak I'm working with a problem in Spivak's Calculus book, Chapter 8, which is stated as follows:

I think I proved part a) , my proof goes:
Let $y$ be another point in $[a,b]$ and suppose f reaches it's maximum at it, so $f(y)>f(x)$ for all $x$ in $[a,b]$, in particular $f(y)>f(a)$, but we must have $y > a$ so we contradict the fact that a is not a shadow point
Next I have to prove that this leads to a contradiction. My idea was that if $f(a) > f(b)$ then obviously the 'rays' of the sun would hit some points near a, contradicting that those points are shadow points. So what i did, was similar to what a previous user did:
Rising Sun Lemma proof from Spivak.
Since f is continuous on $R$, it is continuous at a, and so $\exists\delta>0\ \forall x \ |x-a|<\delta\implies f(x)>f(b).$ Now, as he pointed out, this cannot be possible, at least at an intuitive level, because a shadow point cannot have 
$f(x)>f(b)$
But as it was pointed out in that post, you must prove this in order to reach a contradiction, because "it could be that for every point $y\in (a,b)$ there is a point $z\in (y,b)$ such that $f(z)>f(y)$. "
Can you give me ideas on how to prove this? I know it has to do with a being the max point, but I just can't get it. Thanks for advance.
 A: To show that $\forall x \in (a, b) , f(x) < f(b)$, pick any $x \in (a, b)$ and suppose that $f(x) \geq f(b)$. Since $b$ is not a shadow point, $\forall y > b, f(b) \geq f(y)$.
By the Extreme Value Theorem, $\exists m \in [x, b], \forall z \in [x, b],  f(m) \geq f(z)$. Since $x$ is a shadow point, there should exist some point $y \in [x, b]$ (it cannot be to the right of $b$ by the logic above) with $f(y) > f(x)$, in particular, $f(m) \geq f(y) > f(x) \geq f(b)$. So $m \neq b$, which means that $m \in (x, b)$, i.e. is a shadow point.
But since $m$ is a shadow point, there should be some point $p > m$ with $f(p) > f(m)$, which is not the case, so $m$ is not a shadow point, contradiction.  
Now, we know that $\forall x \in (a, b) , f(x) < f(b)$, and can combine this with part a) to end the problem. 
A: There are some issues in the proof for $(a)$. For instance, if $y$ is the point in the interval $[a,b]$ such that $f$ reaches its maximum value in there, then $f(y)$ is greater or equal to $f(x)$, for all $x\in[a,b]$. In another case, we have that
$$\forall x \in [a,b] \ \  f(y) > f(x) \quad \textrm{implies} \quad f(y)>f(y)$$
you see? Also, nowhere in the proof do you use that $f(a)>f(b)$, which suggests that there is something wrong.
