I am trying to prove that the shortest distance between two points is a straight line. And it is almost complete, if I can use the following theorem.

Assume,

$\displaystyle f(x),g(x)$ are Integratable on $\displaystyle [a,b]$.

And,

$\displaystyle \int_a^b [f(x)]^2 dx\leq \int_a^b [g(x)]^2 dx$

Then is it true that,

$\displaystyle \int_a^b |f(x)|dx\leq \int_a^b |g(x)|dx$?

And one more,

$\displaystyle 0\leq \int_a^b f(x)dx\leq \int_a^b g(x)dx$

Und,

$\displaystyle 0\leq \int_a^b a(x)dx\leq \int_a^b b(x)dx$

Then,

$\displaystyle 0\leq \int_a^b f(x)a(x)dx\leq \int_a^b g(x)b(x)dx$

I tried the Cauchy-Swarthz inequality because this is an inner product space but it does not help sufficiently.