Is there is any way to prove the Faraday's law of induction? I want to know if there is any way to prove the Faraday's law or is it just an experimental observed phenomena?
More specifically, is there any reason why the proportionality constant is 1? How did Faraday discover it? I also heard that we can prove Faraday's law using the principle of least action, but is that true?
 A: Every proof starts with axioms — things that are merely assumed rather than proven. The axioms should be motivated by their applicability to a broad range of phenomena, but utlimately they are just axioms. We test them by comparing their predictions to experiment. Axioms that apply to a broader variety of experiments are better.
If we start with quantum electrodynamics (which has very broad applicability), then one of the axioms is that the components of the electric and magnetic fields are all encoded in a single gauge field $A_a$, where the index $a$ takes values in $\{0,1,2,3\}$. These four index-values correspond to the four dimensions of spacetime (0 for the time dimension, and 1,2,3 for the space dimesions). 
To describe how the electric and magnetic fields are encoded in the gauge field, first define
$$
 F_{ab}\equiv \partial_a A_b - \partial_b A_a
\tag{1}
$$
where $\partial_a$ denotes the partial derivative with respect to the $a$-th coordinate in spacetime. (The gauge field is a function of all four of these coordinates.) This quantity is antisymmetric, $F_{ab}=-F_{ba}$, so it has six independent components. The three components $F_{ab}$ with $b=0$ are the components of the electric field $E_a$, and the three components $F_{ab}$ with $a,b\neq 0$ are the components of the magnetic field $B_{ab}=-B_{ba}$, which are usually written with a single index like this:
$$
   B_1 \equiv B_{23}
 \hskip2cm
   B_2 \equiv B_{31}
 \hskip2cm
   B_3 \equiv B_{12}.
\tag{2}
$$
(Notice the cyclic pattern.)
With these identifications, equation (1) implies Faraday's law. To see this, first notice that equation (1) implies
$$
 \partial_a F_{bc} + \partial_b F_{ca} + \partial_c F_{ab} = 0.
\tag{3}
$$
(Again, notice the cyclic pattern.) The components of equation (3) with all indices non-zero say that the divergence of the magnetic field is zero. The components of equation (3) with one index equal to zero, say $a=0$, give Faraday's law:
$$
 \partial_0 B_{bc} + \partial_b E_{c} - \partial_c E_{b} = 0.
\tag{4}
$$
(Disclaimer: I didn't check to see if my sign-conventions for $E$ and $B$ are standard, but if they're not, then just reverse all of the signs in equation (2). It's just a convention.)

I also heard that we can prove it using the principle of least action is that true?

Formulating things in terms of the gauge field allows the equations of electrodynamics (classical or quantum) to be expressed using the action principle. 
