When does the Wasserstein metric attain inequality WLOG? I’m reading a classic paper [1] that describes a version of the Wasserstein metric (aka Mallows metric), defined as follows. Let $F$ and $G$ be probabilities in $\mathbb{R} ^B$, and let $U \sim F$ and $V \sim G$ be $B$-valued RVs with marginal distributions $F$ and $G$ and an arbitrary joint distribution. Then:
$$d_p(F,G) := \inf\limits_{ \substack{U \sim F \\ V \sim G} } E\Big[  || U-V||^p \Big]^{1/p}$$
The paper says the infimum is always attained for some joint distribution of $(U,V)$ and then goes on to prove various results by considering only the case in which the infimum is attained, "without loss of generality". For example (Lemma 8.6):

In view of Lemma 8.1 [stating that the infimum is attained and that $d_2(.,.)$ is a metric], assume without loss of generality that $(U, V)$ are independent and $$d_p(F, G) = E\Big[  || U-V||^p \Big]^{1/p}$$

First, I don't understand why this holds WLOG. Second, I'm confused about why the infimum would be attained for independent $(U,V)$: intuitively, wouldn't opposite be true? I would expect the "distance" to be smallest when $(U,V)$ are as correlated as their marginals would allow.
References
[1] Bickel, P. J., & Freedman, D. A. (1981). Some asymptotic theory for the bootstrap. The Annals of Statistics, 1196-1217.
 A: This issue basically has to do with how the proofs are structured; I think you've slightly misunderstood how the assumptions work with the proof itself.  Lemma 8.6 is just a proof that the triangle inequality holds for pairs of the form ($\sum U_i, \sum V_i)$ and the distance metric as defined at the top of Section 8: Mathematical Appendix:

where $B$ is a suitably-defined Banach space.  
B&F start out by stating that in view of Lemma 8.1, they can assume that the infimum is attained (implicitly, for the function $E\{ ||\cdot||^p\}^{1/p}$ between $U_i$, $V_i$ and $\sum U_i$, $\sum V_i$), and there's no loss of generality in doing so, because 8.1 shows that the infimum is attained.  This fact is used in 8.1 in the proof that $d_p$ is a metric on $\Gamma_p$, i.e., part a is used to prove part b, so, by making that assumption, they can use the fact that $d_p$ is a metric in subsequent steps / lemmas.
ETA for the first point:
Note that the distance is actually between the laws $\alpha$ and $\beta$, not between specific random variables $X$ and $Y$.  The distance between $\alpha$ and $\beta$ is defined as the infimum of the given function over all $X$ and $Y$ such that: a) $X$ and $Y$ are $B$-valued random variables where b) $X$ has law $\alpha$ and $Y$ has law $\beta$.  The point of the proof of Lemma 8.1 is that a pair of such r.v.s exist that attain the infimum, but the distance isn't between that pair, or any other pair for that matter, it's between the laws $\alpha$ and $\beta$.  
With respect to your second question about why the infimum would be attained for independent ($U, V$) - I think you've slightly misunderstood the use of the independence argument.  The attainment of the infimum has nothing to do with orthogonality between $U_j, V_j$, as the proof of Lemma 8.1 makes no such assumption, and indeed nor does Lemma 8.6.  The orthogonality assumption is across the pairs $(U_j, V_j)$ ("let us assume without loss of generality that the pairs $(U_j, V_j)$ are independent") and is not used in the proof that the infimum is achieved (Lemma 8.1), but rather in the proof that the triangle inequality holds (Lemma 8.6).  
ETA for the second point:
Now, when B&F write out 8.6, they refer to $(U_i, V_i)$ rather than $\alpha$ and $\beta$, which can certainly cause confusion.  But they do so because they are referring to the function $E\{ ||U-V||^p\}^{1/p}$, and doing so allows them to use Minkowski in their proof.  In general, that function is not equal to $d_p(\alpha, \beta)$, so the proof isn't helpful... but if we happen to choose the $U_i, V_i$ pair that achieve the infimum in Lemma 8.1, the function  is equal to $d_p(\alpha, \beta)$, by definition of the distance. Therefore, the proof using Minkowski can be transformed from a proof about $E\{ ||X-Y||^p\}^{1/p}$ to a proof about $d_p(\alpha, \beta)$, and the properties about $E\{ ||X-Y||^p\}^{1/p}$ which they just proved also hold for $d_p(\alpha, \beta)$, which is what we are really interested in.
