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 ] $

By assumption, $ J $ is lower semicontinuous at $ \hat{ y } $:


 * $ \displaystyle \forall \epsilon_1 > 0 : \forall n \in \N : \exists N \in \N : \left ( { n > N } \right ) \implies \left ( { J \left [ { \hat{ y } } \right ] - J \left [ { y_n } \right ] < \epsilon_1 } \right ) $

Hence, for sufficiently large $ n $:


 * $ \displaystyle \inf_y J \left [ { y } \right ] \le J \left [ { \hat{ y } } \right ] < J \left [ { y_n } \right ] + \epsilon_1 $

By the definition of minimizing sequence, where the label $ n $ has been replaced with $ m $:


 * $ \displaystyle \forall \epsilon_2 : \forall m \in \N : \exists M \in \N : \left ( { m > M } \right ) \implies \left ( { \left \vert { J \left [ { y_m } \right ] - \inf_y J \left [ { y } \right ] } \right \vert < \epsilon_2 } \right ) $

From this, for sufficiently large $ m $:


 * $ \displaystyle J \left [ { y_m } \right ] - \epsilon_2 < \inf_y J \left [ { y } \right ] $

Then:


 * $ J \left [ { y_m } \right ] - \epsilon_2 < J \left [ { \hat{ y } } \right ] < J \left [ { y_n } \right ] +\epsilon_1 $

Here $ \epsilon_1$, $ n $ have similar properties like $ \epsilon_2$, $ m $, but are otherwise arbitrary and independent.

Let $ \epsilon_1 = \epsilon_2 = \epsilon $, $ n = m $, $ M = N $. Arbitrariness is still not affected.

Then:


 * $ J \left [ { y_n } \right ] - \epsilon < J \left [ { \hat{ y } } \right ] < J \left [ { y_n } \right ] +\epsilon $

Subtract $ J \left [ { y_n } \right ] $ from all the terms.

This results into:


 * $- \epsilon < J \left [ { \hat{ y } } \right ] - J \left [ { y_n } \right ] < \epsilon $

or


 * $\left \vert { J \left [ { y_n } \right ] - J \left [ { \hat{ y } } \right ] } \right \vert < \epsilon $

Therefore this relation inherits all the constraints on its values $ n $, $ N $, $ \epsilon$, and by definition is a limit:


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