Bolzano-Weierstrass proof correction 
Bolzano-Weierstrass theorem Every sequence $\{ x_n \}_{n=1}^\infty$ bounded in $\mathbb R$ has, at least, a convergent subsequence.

Proof Since $\{ x_n \}_{n=1}^\infty$ is bounded, then $\exists a,b\in \mathbb R$, $a< b$, so $\{ x_n \}_{n=1}^\infty \subset [a,b]$.
If $\{x_n\}_{n=1}^\infty$ is a finite set, then it has a constant subsequence, which is convergent.
If $\{x_n\}_{n=1}^\infty$ is an infinite set, because of the Bolzano-Weierstrass theorem, there exists an accumulation point $x_0$ of $\{x_n\}_{n=1}^\infty$ (already proved).
Let's build a subsequence of $\{x_n\}_{n=1}^\infty$ that converges to $x_0$: since
$\forall \varepsilon > 0 \;\, B(x_0, \varepsilon)\setminus\{x_0\}\cap
\{x_n\}_{n=1}^\infty \neq \emptyset$ (because of the definition of accumulation point), let $\varepsilon_1 =1$, then $\exists
n_1 \in \mathbb{N} $ s.t. $ x_{n_1} \in B(x_0, \varepsilon_1)$.
Let $m_2 \in \mathbb{N}$,
$\varepsilon_2 = \frac{1}{m_2} < d(x_{n_1}, x_0)$, then $\exists n_2\in \mathbb{N}
\;\, (n_2 > n_1)$ s.t. $ x_{n_2} \in B(x_0, \varepsilon_2)$.
Inductively we have that $\{x_{n_k}\}_{k=1}^\infty$ is a subsequence of $\{x_n\}_{n=1}^\infty$ and
$n_k < n_{k+1}$, so $ d(x_{n_k}, x_0) <
\varepsilon_k = \frac{1}{m_k}$.
Then $\lim_{k\to\infty} d(x_{n_k}, x_0) = 0$, which is equivalent to $\lim_{k\to\infty} x_{n_k}= x_0$ so $\{x_{n_k}\}_{k=1}^\infty$ is a convergent subsequence.
Question How can I rewrite the last part (the inductive part seems unclear to me) and everything unclear, since I can't understand why $\forall \varepsilon > 0 \;\, \exists k_0 \;\, \forall k \geq k_0 \;\, d(x_{n_k}, x_0) < \varepsilon $ (def. of convergence)...? Why is built $\{m_k\}$?
 A: The inductive part is unecessarily complicated. Here is a simpler version:
For $\epsilon=1$, $\exists n_1 \in \mathbb{N} $ s.t. $ x_{n_1} \in B(x_0, 1)$.
For $\epsilon=\frac12$, $\exists n_2\in \mathbb{N}
\;\, (n_2 > n_1)$ s.t. $ x_{n_2} \in B(x_0, \frac12)$
Having defined $n_k$ so that $n_k>n_{k-1}$ and $ x_{n_k} \in B(x_0, \frac1k)$ define $n_{k+1}$ as follows:
For $\epsilon=\frac1{k+1}>0$, $\exists n_{k+1}\in \mathbb{N}
\;\, (n_{k+1} > n_k)$ s.t. $ x_{n_{k+1}} \in B(x_0, \frac1{k+1})$
The subsequence $(x_{n_k})$ satisfies $ x_{n_k} \in B(x_0, \frac1k)$ or equivalently
$$ d(x_{n_k}, x_0) <\frac{1}{k}$$
By the squeeze theorem,
 $$ d(x_{n_k}, x_0) \to 0$$
as $k\to +\infty$ and the proof is complete!
A: You are aiming to find a subsequence that converges to $x_0$. To do this, you are showing that  given $\epsilon$>0, $\exists$ k$\epsilon$$\mathbb{N}$ such that B($x_0$;$\epsilon$) contains all terms in the subsequence that appear after the $k^{th}$ term. Now read the proof again with this in mind and let me know if you are still unable.         
