Spivak 11 Appendix Question 8 understanding solution I need help understanding part of the solution to Chap 11 Appendix on Convexity, Question 8.  I don't believe its necessary to see the whole question or solution to understand my issue.  We have the inequality for convexity where $0<t<1$.  The solution states:

I don't understand the part where it says if strict inequality for one $t$ then using the weak inequality, we can prove that strict inequality holds for all $t$.
 A: Geometrically, the weak inequality states that the graph of $f$ between any two points $(x,f(x))$ and $(y, f(y))$ cannot lie above the line segment connecting those two points. 
In the diagram let's label the left and right dots as $A:=(x,f(x))$ and $B:=(y, f(y))$. The dashed segment $AB$ joining $A$ and $B$ represents the points with vertical coordinates
$$tf(x) + (1-t)f(y)$$
as $t$ varies between $0$ and $1$.
The middle dot (label it $C$) represents the value of $t$ where there is strict inequality. By strict inequality at any specific value of $t$, the dashed segment $AB$ lies strictly above the solid segments $AC$ and $CB$ for every $t$ between $0$ and $1$, as illustrated in the diagram. In turn, the graph of $f$ cannot lie above either of the two solid segments (we are applying the weak inequality twice). These two facts imply that the graph of $f$ lies strictly below the dashed segment $AB$ for every $t$ between $0$ and $1$. This is equivalent to saying that strict inequality holds for all $t$ between $0$ and $1$.
TL;DR:
$$
\begin{align}
f(tx + (1-t)y) &= \text{height of graph of $f$}\\& \le \text{height of solid segment} \\&< \text{height of dashed segment}\\&=tf(x) + (1-t)f(y)
\end{align}$$
