How to understand minimising length is equivalent to energy minimising?

Question

How to understand the equivalent above red line in picture below ?

Answer 1

Here is an idea of a proof:

For any curve we can choose a reparametrisation such that its speed is constant, such that we actually have $L(\gamma)^2 = 2 E(\gamma)$ (this follows from going through the Cauchy-Schwarz inequality that already underlies the inequality in the text) and we only have to consider curves with constant speed. Now it is clear that if any of those curves minimizes $E$ it also minimizes $L$ and since $L$ is always positive the converse is also clear.

So I would say that the text is somewhat misleading in saying that they have provided all the necessary ingredients to prove this statement, but it is true.