Definitions
Weak Containment
Let $G$ be a Lie group and $\sigma:G\to \mathcal U(\mathcal H)$ and $\pi:G\to \mathcal U(\mathcal K)$ be two (continuous) unitary representations of $G$.
We say that $\sigma$ is weakly contained in $\pi$ if for all compact subsets $C$ of $G$, for all $\epsilon>0$, and for unit vectors $u_1, \ldots, u_n$ in $\mathcal H$, there exist unit vectors $v_1, \ldots, v_n\in \mathcal K$ such that $$|\langle \sigma(g)u_i, u_j\rangle - \langle \pi(g)v_i, v_j \rangle|< \epsilon$$ for all $g\in C$, and all $1\leq i, j \leq n$.
Induced Representation
Let $\Gamma$ be a lattice in a Lie group $G$ and let $\pi:\Gamma\to \mathcal U(\mathcal H)$ be a unitary representation of $\Gamma$. Using $\pi$, we can construct a unitary representation of $G$ as follows.
Let $L_\Gamma(G;\mathcal H)$ be the set of all measurable functions $\phi:G\to \mathcal H$ such that $$\phi(g\gamma^{-1}) = \pi(\gamma)\phi(g)$$ for all $g\in G$, $\gamma\in \Gamma$. (We identify two maps if they agree almost everywhere).
Fix $\phi\in L_\Gamma(G;\mathcal H)$. Note that $\|\phi(g\gamma)\|_{\mathcal H} = \|\phi(g)\|_{\mathcal H}$ for all $\gamma\in \Gamma$ and all $g\in G$. Thus $g\mapsto \|\phi(g)\|_{\mathcal H}:G\to \mathbb R$ factors through $G/\Gamma$ and allows us to define the $L^2$-norm of $\phi$ by writing $$\|\phi\|_2^2:= \int_{G/\Gamma} \|\phi(g)\|^2_{\mathcal H} dg$$
Let $L^2_\Gamma(G;\mathcal H)$ denote the set of all members of $L_\Gamma(G;\mathcal H)$ with finite $L^2$-norm. This is a Hilbert space. Also, define an action of $G$ on $L^2_\Gamma(G, \mathcal H)$ by writing $$(g\cdot \phi)(x) = \phi(g^{-1}x)$$ This gives a unitary representation of $G$ on $L^2(G;\mathcal H)$, and this representation is called the induced representation of $G$ coming from $\pi$. We denote it by $\text{Ind}_\Gamma^G(\pi)$.
Question
Let $G$ be a Lie group and $\sigma:\Gamma\to \mathcal U(\mathcal H)$ and $\pi:\Gamma\to \mathcal U(\mathcal K)$ be two unnitary representations of $\Gamma$. I want to show that
If $\sigma$ is weakly contained in $\pi$, then $\text{Ind}_\Gamma^G(\sigma)$ is weakly contained in $\text{Ind}_\Gamma^G(\pi)$.
I am completely stuck here. It is not even clear to me how to proceed if we take a singleton in place of a compact set $C$.