Consider the following theorem in Murphy's book "$C^*$-algebras and operator theory":
Why do we need to truncate the net in order to conclude that $(u_\lambda)_{\lambda}$ is bounded below?
Would the following be correct? Fix $\lambda_0 \in \Lambda$ and consider $\Lambda':= \{\lambda \in \Lambda: \lambda \geq \lambda_0\}$. Then by definition of increasing net we have $u_{\lambda_0} \leq u_\lambda$ for all $\lambda \in \Lambda'$ so $(u_{\lambda})_{\lambda \in \Lambda'}$ is bounded below by $u_{\lambda_0}$. Moreover, if we can show that $(u_\lambda)_{\lambda \in \Lambda'} $ converges strongly to $u$, then $(u_\lambda)_{\lambda \in \Lambda}$ converges strongly to $u$ as well and thus we can safely replace $(u_\lambda)_{\lambda\in \Lambda}$ by $(u_\lambda)_{\lambda\in \Lambda'}$.
One of the key differences between sequences and nets is that there is no first element. For example, you might have $\Lambda = \mathbb{R}$. However, to perform this argument, you need a starting point $\lambda_0$, so you just choose one. Your argument after that is fine.