Let $\Gamma$ be a discrete and countable Group and let $\pi:\Gamma\to \mathcal{B(H)}$ be a unitary representation.
We say that $\pi$ almost has invariant vectors if for every compact (=finite) subset $F\subseteq\Gamma$ and $\varepsilon >0$ there there exists a unit vector $\xi\in\mathcal{H}$ such that $$ \|\pi(s)\xi -\xi\|<\varepsilon\; : \forall s\in F.$$
This is demonstrated to be equivalent to the weak containment of the trivial representation, denoted by $1_\Gamma$, in $\pi$ (see for example Cor. F.1.5, [1], general case of locally compact groups).
In other literature (for example Thm. A.5. and A.12, [2]) I have found the following definition: $(\pi,\mathcal{H})$ admits almost invariant vectors if there exists a sequence of unit vectors $(\xi_n)_n\subseteq \mathcal{H}$ such that $$\|\pi(s)\xi_n-\xi_n\| \to 0 \; : \; \forall s \in \Gamma$$ which is also claimed to be equivalent to the trivial representation being weakly contained in $\pi$.
Are these conditions really equivalent? If so, how would I go about understanding this equivalence? At this point I don't see how these two could be equivalent, without considering nets instead of sequences in the second condition.
Many thanks in advance.
[1] "Kazhdan’s Property (T)" by B. Bekka, P. de la Harpe and A. Valette)
[2] "Amenability of discrete groups by examples" by Kate Juschenko
The implication "$(\pi, H)$ admits almost invariant vectors" $\Rightarrow$ "$(\pi,H)$ almost admits invariant vectors" is relatively straightforward: Choose a sequence $(\xi_n)$ in $H$ with $\|\pi(s)\xi_n-\xi_n\|\to 0$ for all $s\in\Gamma$. If $F\subset\Gamma$ is finite and $\varepsilon>0$ is given, then for each $s\in F$ there is some $N_s$ such that $\|\pi(s)\xi_n-\xi_n\|<\varepsilon$ for $n\geq N_s$. Now choose $n\geq\max\{N_s:s\in F\}$, and put $\xi=\xi_n$.
For the other direction, let $(F_n)$ be an increasing sequence of finite subsets of $\Gamma$ such that $\cup_nF_n=\Gamma$. For each $n$, there exists $\xi_n\in H$ with $\|\pi(s)\xi_n-\xi_n\|<\frac{1}{n}$ for all $s\in F_n$. Then $(\xi_n)$ is your sequence for the criterion of almost invariant vectors.