Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}_n$ be a sequence of simple functions such that: $$ f_n1_{\{|f_n|\leq k\}}\overset{\sigma(L^2,L^2)}{\underset{n}{\longrightarrow}} u_k\qquad\forall k\geq 1 \qquad\text{ et }\qquad \|u_k\|_2\leq 2\|f_n1_{\{|f_n|\leq k\}}\|_2,\qquad \forall n\geq k. $$ Put $\epsilon_k=\frac{1}{2^k}$ $(k\geq 1)$. Why for each $k\geq 1$, there exists a simple function $v_k$ such that: $$ \|u_k-v_k\|_2\leq \min \bigg(\inf_{n\geq k}{\|f_n1_{\{|f_n|\leq k\}}\|_2},\frac{\epsilon_{k-1}}{4k}\bigg) $$

Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}_n$ be a sequence of simple functions such that: \begin{align*} f_n1_{\{\lvert f_n\rvert\leq k\}}\overset{\sigma(L^2,L^2)}{\underset{n}{\longrightarrow}} u_k, &\qquad\forall k\geq 1 \\ \|u_k\|_2\leq 2\|f_n1_{\{\lvert f_n\rvert\leq k\}}\|_2,&\qquad \forall n\geq k. \end{align*} Put $\epsilon_k=\frac{1}{2^k}$ $(k\geq 1)$. Why does there exist, for each $k\geq 1$, a simple function $v_k$ such that: $$ \|u_k-v_k\|_2\leq \min \bigg(\inf_{n\geq k}{\|f_n1_{\{\lvert f_n\rvert\leq k\}}\|_2},\frac{\epsilon_{k-1}}{4k}\bigg) $$