Equivalence of Tightness of Seqeuence of CDFs In Durrett, a sequence of cdf $\{F_n\}$ is called tight if for all $\epsilon > 0$< there exists $M_\epsilon$ such that $\limsup 1-F_n(M_\epsilon)+F_n(-M_\epsilon) \leq \epsilon$.
In Rosenthal, a sequence of probability measure is called tight if for all $\epsilon$, there exists $a<b$ both finite such that $\mu_n([a,b]) \geq 1-\epsilon$ for all $n$.


Are these two definitions equivalent?
I can write $\limsup 1-F_n(M_\epsilon)+F_n(-M_\epsilon) \leq \epsilon$ as $\limsup F_n(-M_\epsilon)-F_n(M_\epsilon) \leq \epsilon-1$, which can be writen as $\liminf \mu_n(-M_\epsilon, M_\epsilon])=\liminf F_n(M_\epsilon)-F_n(-M_\epsilon) \geq 1-\epsilon$
From this, it's clear that the definition in Rosenthal implies the definition in Durrett.
However, the converse is not clear to me. Since we can have for all $n$, $\mu((-M_\epsilon, M_\epsilon]) < 1-\epsilon$ while $\liminf \mu((-M_\epsilon, M_\epsilon]) = 1-\epsilon$, in which case the definition in Rosenthal wouldn't hold.
Thank you.

Assuming Durrett's definition, choose $M$ so that for all $\liminf \mu_n([-M,M]) \geq \frac{\epsilon}{2}$ for a fixed $\epsilon > 0$.
Then eventually, $a_n = \inf_{m \geq n} \mu_m([-M,M]) \geq 1-\epsilon$ since $\liminf \mu_n([-M,M]) > 1-\epsilon$.
So eventually, $\mu_n([-M,M]) \geq 1-\epsilon$, and this suffices since finite collection of probability measures is tight.
