Let $E$ be a real normed vector space et $(u_n)_{n\ge0}$, $(v_n)_{n\ge0}$ be convergent sequences of vectors.
Suppose $u_n$ and $v_n$ are linearly dependent for every $n\in\mathbb{N}$.
I would like to prove that the vectors $a=\lim u$ and $b=\lim v$ are also linearly dependent.
I know how to prove this (see below), but I have the feeling that I miss something easier ... Comments would be appreciated !
For all $n\in\mathbb{N}$, there exists $(\lambda_n,\mu_n)\in\mathbb{R}^2-\{(0,0)\}$ such that $\lambda_nu_n+\mu_nv_n=0$
Consider the sequences $\alpha$ and $\beta$ defined by :
$$\forall n\in\mathbb{N},\quad\alpha_n=\frac{\lambda_n}{\sqrt{\lambda_n^2+\mu_n^2}}\quad\mathrm{and}\quad\beta_n=\frac{\mu_n}{\sqrt{\lambda_n^2+\mu_n^2}}$$
$[-1,1]^2$ is compact and we can WLOG suppose than $\alpha$ and $\beta$ converge. Let us denote by $r$, $s$ their limits.
We have $r^2+s^2=\lim_{n\to\infty}(\alpha_n^2+\beta_n^2)=1$, which proves that $(r,s)\neq(0,0)$.
Since $ra+sb=0$, we conclude that $a,b$ are linearly independent.