optimization – Theorem of Weierstrass for Closed Functions

In a lecture, we stated and proved the following theorem of Weierstrass, but I would have a question to it. First, let me give some basic definitions and assumptions before stating it.


Assumptions:

  • $X$ is a finite vector space.
  • Let $J:Xrightarrowmathbb R_{infty}:=mathbb R cup { infty }$ be an extended real function.

Definitions:

  • The effective domain of $J$ is the set of points where $J$ is finite, i.e. $text{dom}(J) := { xin X vert J(x) < infty }.$
  • The function $J: Xmathbb rightarrow R_{infty}$ is said to be proper if $text{dom}(J) ne emptyset$.
  • A function $J: Xrightarrow mathbb R_{infty}$ is closed if its epigraph, $text{epi}(J) := { (x, alpha) in Xtimes mathbb R vert J(x) leqalpha },$ is closed. This is equivalent (by a Theorem) to $J$ being lower semi-continuous, i.e. for any sequence ${x_n}_nsubset X$ with $x_nrightarrow x$ it holds that $J(x^{star}) leq liminf_{ntoinfty}J(x_n)$.

Now we proved: Theorem: Let $J:Xrightarrowmathbb R_{infty}$ be proper, closed and let $Csubset X$ be compact with $C cap text{dom}(J) ne emptyset$. Then $J$ attains its minimal value over $C$. Since $J$ is closed ($Leftrightarrow J$ is lower semi-continuous) $Rightarrow J(x^{star}) leq liminf_{ktoinfty}J(x_{n_k})$

Proof (in our lecture): According to the Bolzano-Weierstrass theorem, there exists a convergent subsequence ${x_{n_k}}_{k}overset{ktoinfty}{rightarrow} x^{*}in C$. Since $J$ is lower semi-continuous $Rightarrow J(x^{star}) leq liminf_{ktoinfty} J(x_{n_k}) = lim_{ktoinfty}J(x_{n_k}) = min_{xin C}J(x)$. Thus, $x^{star} in C$ is the minimizer. QED

Question: I do not really understand why $lim_{ktoinfty}J(x_{n_k}) = min_{xin C}J(x)$, which seems to be a very crucial point.