A simple pendulum of length L with a bob of mass m is deflected...

According to the law of conservation of mechanical energy, the total mechanical energy of the system remains constant since the only force doing work is gravity. The tension in the string does no work as it is always perpendicular to the instantaneous velocity of the bob.

Let the lowest point of the bob's trajectory be the reference level for potential energy. When the bob is released from rest at an initial height $h_i$, its initial kinetic energy is zero. Thus, the total initial energy is purely potential:
$E_i = mgh_i$

As the bob swings and the string encounters the peg, the bob starts moving in a circular path of a smaller radius $(L-x)$. Since the initial height of the bob is less than the height of the peg, the bob will not reach the level of the peg on the other side. This ensures that the string remains taut throughout the motion and does not go slack.

At the maximum height $h_f$ on the other side of the peg, the instantaneous velocity of the bob becomes zero again. The total energy at this point is:
$E_f = mgh_f$

By conservation of mechanical energy, $E_i = E_f$:
$mgh_i = mgh_f$
$\Rightarrow h_i = h_f$

Therefore, the maximum height reached by the bob is exactly equal to its initial height.