Conditions for Limit Function to be Limit Minimizing Function of Functional

Theorem
Let $ y $ be a real function.

Let $ J \left [ { y } \right ] $ be a functional.

Let $ \left \{ { y_n } \right \} $ be a minimizing sequence of $ J $.

Let:


 * $ \displaystyle \lim_{ n \to \infty } y_n = \hat{ y } $

Suppose $ J $ is lower semicontinuous at $ y = \hat{ y } $.

Then:


 * $ \displaystyle J \left [ { \hat{ y } } \right ] = \lim_{ n \to \infty } J \left [ { y_n } \right ] $

Proof
By definition of minimizing sequence:


 * $ \displaystyle \inf_y J \left [ { y } \right ] = \lim_{ n \to \infty } J \left [ { y_n } \right ] $

Any mapping from this sequence either minimises the functional or not.

This is true for the limit mapping as well:


 * $ \displaystyle J \left [ { \hat{ y } } \right ] \ge \inf_y J \left [ { y } \right ] $

Since $ J $ is lower semicontinuous at $ \hat{ y } $:


 * $ \displaystyle \forall \epsilon > 0 : \exists n \in N : J \left [ { \hat{ y } } \right ] - J \left [ { y_n } \right ] < \epsilon $

Take the limit $ n \to \infty $:


 * $ \displaystyle J \left [ { \hat{ y } } \right ] \le \lim_{ n \to \infty } J \left [ { y_n } \right ] + \epsilon $

Hence:


 * $ \displaystyle \inf_y J \left [ { y } \right ] \le J \left [ { \hat{ y } } \right ] \le \lim_{ n \to \infty } J \left [ { y_n } \right ] + \epsilon $

or


 * $ \displaystyle \inf_y J \left [ { y } \right ] \le J \left [ { \hat{ y } } \right ] \le \inf_y J \left [ { y } \right ] + \epsilon $

By Squeeze Theorem:
 * $ \displaystyle J \left [ { \hat{ y } } \right ] = \inf_y J \left [ { y } \right ]$

or


 * $ \displaystyle \lim_{ n \to \infty } J \left [ { y_n } \right ] = J \left [ { \hat{ y } } \right ] $