In the proof of the Rellich Compactness Theorem in "Partial Differential Equations", Evans uses the notation $$\limsup_{j,k\to \infty}\, \lvert \lvert u_{m_j} - u_{m_k} \rvert\rvert_{L^q(V)}\leq \delta,$$ but I'm not sure if I understand it correctly. If $\{x_n\}_n\subset \mathbb{R}$ is a sequence, we have
$$\limsup_{n\to \infty} x_n \leq K\Leftrightarrow \text{For every } \epsilon>0, \text{there exists an } N\in \mathbb{N} \text{ such that } x_n \leq K+\epsilon \text{ for all } n\geq N.$$
So is it true that $$\limsup_{j,k\to \infty}\, \lvert\lvert u_{m_j} - u_{m_k}\rvert\rvert_{L^q(V)}\leq \delta\Leftrightarrow\text{For every } \epsilon > 0, \text{there exists an } N\in \mathbb{N} \text{ such that } \lvert\lvert u_{m_j} - u_{m_k} \rvert\rvert_{L^q(V)}\leq \delta + \epsilon\text{ for all } j,k\geq N?$$
In the last line of the proof, Evans argues that $\limsup_{j,k\to \infty}\, \lvert\lvert u_{m_j} - u_{m_k}\rvert\rvert_{L^q(V)} = 0$ implies that $u_{m_j}$ is a cauchy sequence and therefore converges in $L^q$. This works well with "my" definition, since $$\limsup_{j,k\to \infty}\, \lvert\lvert u_{m_j} - u_{m_k}\rvert\rvert_{L^q(V)}=0 \Leftrightarrow \text{For every } \epsilon > 0,\text{ there exists an } N\in \mathbb{N} \text{ such that } \lvert\lvert u_{m_j} - u_{m_k}\rvert\rvert_{L^q(V)}< \epsilon \text{ for all } j,k\geq N \Leftrightarrow u_{m_j} \text{ is a cauchy sequence}.$$