Two equivalent definitions Riemann integral Definition 7.1 in Apostol's book gives the following definition for the Riemann integral. Let $P=\{x_0,x_1,\ldots,x_n\}$ be a partition of $[a,b]$ and $t_k$ any point in $[x_{k-1},x_k]$. Denote $S(P,f)=\sum_{k=1}^n f(t_k)(x_k-x_{k-1})$. 
Definition 1: We say that $f$ is Riemann integrable on $[a,b]$ if there exists a real number $A$ such that, for all $\epsilon>0$, there exists a partition $P_\epsilon$ of $[a,b]$ such that, for each finer partition $P$ and for each arbitrary choice $t_k\in [x_{k-1},x_k]$, it holds $|S(P,f)-A|<\epsilon$.
One of the intuitions I have to better understand the Riemann integral could be summarized in the following definition:
Definition 2: We say that $f$ is Riemann integrable on $[a,b]$ if there exists a real number $A$ and a sequence of partitions $\{P_n\}$ with mesh tending to $0$, such that, for each arbitrary choice $t_k\in [x_{k-1},x_k]$ in $P_n$, it holds $\lim_{n\rightarrow\infty} S(P_n,f)=A$.
I have never seen a proof of the equivalence of both definitions. Maybe they are not equivalent.
 A: The definitions are equivalent.
We have the Riemann criterion whereby a function $f$ is integrable over $[a,b]$ according to Definition 1 if and only if for every $\epsilon > 0$ there exists a partition $P$ such that upper and lower sums satisfy $U(P,f) - L(P,f) < \epsilon$.
Suppose $f$ satisfies Definition 2.  Given $\epsilon > 0$ there exists a positive integer $N$ such that for all $n \geqslant N$ and any choice of tags $\{t_j\}$ we have $|S(P_n,f, \{t_j\}) - A| < \epsilon/4$. In particular, we have
$$\tag{*} A - \epsilon/4 < S(P_N,f,\{t_j\}) < A + \epsilon/4.$$
Consider any subinterval $I_j = [x_{j-1},x_j]$ of $P_N$.  Let $m_j = \inf_{x \in I_j} f(x)$ and $M_j = \sup_{x \in I_j}f(x)$.  
There exist points $\alpha_j, \beta_j \in I_j$ such that
$$m_j \leqslant f(\alpha_j) < m_j + \frac{\epsilon}{4(b-a)}, \\ M_j -  \frac{\epsilon}{4(b-a)} < f(\beta_j) \leqslant M_j. $$
Multiplying by $(x_j - x_{j-1})$, summing over $j$ and using (*) we get
$$A - \epsilon/4<S(P_N,f, \{\alpha_j\})< L(P_N,f) + \epsilon/4, \\ U(P_N,f) - \epsilon/4 < S(P_N,f, \{\beta_j\})< A + \epsilon/4$$
This implies $A-  \epsilon/2 < L(P_N,f)$ and  $U(P_N,f) < A + \epsilon/2$.
and, hence, 
$$U(P_N,f) - L(P_N,f) < \epsilon.$$
Since we are able to find a partition where the Riemann criterion is satisfied, a function that is integrable under Definition 2 is also integrable under Definition 1.
The converse is easy to prove.
Addendum 
There are two possible interpretations of $\lim_{n \to \infty} S(P_n,f) = A$ in Definition 2.
(2a)   For every $\epsilon > 0$ there exists a positive integer $N(\epsilon)$ such that if $n \geqslant N(\epsilon)$, then $|S(P_nf, \{t_j\})-A| < \epsilon$ holds for every choice of tags $\{t_j\}$.
(2b)   For every $\epsilon > 0$ and each choice of tags $\{t_j\}$, there exists a positive integer $N(\epsilon, \{t_j\})$ such that if $n \geqslant N(\epsilon, \{t_j\})$, then $|S(P_nf, \{t_j\})-A| < \epsilon$ holds.
To be precise, Definition 1 is equivalent to Definition 2a.  It was shown above that (2a) implies (1).  The converse follows because if a function is Riemann integrable under Definition 1, there is an equivalence to the statement that for any $\epsilon > 0$ there exists $\delta > 0$ such that $\|P\| < \delta $ implies that $|S(P,f, \{t_j\}) - A| < \epsilon$ for any choice of tags.
