絶対収束
≤ ‖ ∑ i = 1 N ε a i − A ‖ + ‖ ∑ i ∈ I σ , ε a σ ( i ) ‖ ≤ ‖ ∑ i = 1 N ε a i − A ‖ + ∑ i ∈ I σ , ε ‖ a σ ( i ) ‖ {\displaystyle \leq \left\|\sum _{i=1}^{N_{\varepsilon }}a_{i}-A\right\|+\left\|\sum _{i\in I_{\sigma ,\varepsilon }}a_{\sigma (i)}\right\|\leq \left\|\sum _{i=1}^{N_{\varepsilon }}a_{i}-A\right\|+\sum _{i\in I_{\sigma ,\varepsilon }}\|a_{\sigma (i)}\|} ≤ ‖ ∑ i = 1 N ε a i − A ‖ + ∑ i = min { σ ( k ) | k ∈ I σ , ε } max { σ ( k ) | k ∈ I σ , ε } ‖ a i ‖ ≤ ‖ ∑ i = 1 N ε a i − A ‖ + ∑ i = N ε + 1 ∞ ‖ a i ‖ ≤ ε {\displaystyle \leq \left\|\sum _{i=1}^{N_{\varepsilon }}a_{i}-A\right\|+\sum _{i=\min\{\sigma (k)|k\in I_{\sigma ,\varepsilon }\}}^{\max\{\sigma (k)|k\in I_{\sigma ,\varepsilon }\}}\|a_{i}\|\leq \left\|\sum _{i=1}^{N_{\varepsilon }}a_{i}-A\right\|+\sum _{i=N_{\varepsilon }+1}^{\infty }\|a_{i}\|\leq \varepsilon }
すなわち
∀ ε > 0 , ∃ M σ , ε ∀ N > M σ , ε , ‖ ∑ i = 1 N a σ ( i ) − A ‖ < ε {\displaystyle \forall \varepsilon >0\ ,\exists M_{\sigma ,\varepsilon }\,\forall N>M_{\sigma ,\varepsilon }\,\,,\left\|\sum \limits _{i=1}^{N}a_{\sigma (i)}-A\right\|<\varepsilon }