Conditions for Limit Function to be Limit Minimizing Function of Functional

Theorem
Let $y$ be a real function.

Let $J\sqbrk y$ be a functional.

Let $\lbrace {y_n}$ 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:


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

Proof
By definition of minimizing sequence:


 * $\displaystyle\inf_y J\sqbrk y=\lim_{n\to\infty} J\sqbrk {y_n}$

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

This is true for the limit mapping as well:


 * $\displaystyle J\sqbrk {\hat y}\ge\inf_y J\sqbrk y$

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


 * $\displaystyle\forall\epsilon_1>0:\forall n\in\N:\exists N\in\N:\paren {n>N} \implies\paren {J\sqbrk {\hat y}-J\sqbrk {y_n}<\epsilon_1}$

Hence, for sufficiently large $n$:


 * $\displaystyle\inf_y J\sqbrk y \le J\sqbrk {\hat y}M} \implies\paren {\size {J\sqbrk {y_m}-\inf_y J\sqbrk y}<\epsilon_2}$

From this, for sufficiently large $m$:


 * $\displaystyle J\sqbrk {y_m}-\epsilon_2<\inf_y J\sqbrk y$

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 ] $