Let $\mathcal{L}(\mathbb{R}^m,\;\mathbb{R}^k):=\{T:\mathbb{R}^m \longrightarrow \mathbb{R}^k : T$ is a linear transformation$\}$ the Vector Space of Linear Transformations with norm $\|T||=\sup\{\|Tx\|:\|x\|=1\}$.
Let $(T_n)_{n\ge1}$ a sequence in $\mathcal{L}(\mathbb{R}^m,\;\mathbb{R}^k)$ and $(u_n)_{n \ge 1}$ $\subset$ $\mathbb{R}^m$ such that :
$\forall n\ge 1, \|u_n\|=1$ and $u_n \to u$ (convergent)
$\forall n\ge 1, \|T_n\|\ge 1$
Then we can conclude that: $\liminf_{n \to \infty}\|T_n(u_n)\|=L>0$.
I think the proof is by contradiction, suppose $\liminf_{n \to \infty}\|T_n(u_n)\|=0$, but how to use the condition $\|T_n\|\ge 1$?
Any hints would be appreciated.