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.