Preserved linear independence of vectors

I have been considering the following problem in Multivariable Calculus:

Consider linearly independent vectors $a_1,$$a_2 ... a_k$ in $\mathbb{R}^n$. Show that there exists $\delta >0$ such that if $b_1,$$b_2,$$... b_k$ in $\mathbb{R}^n$ satisfy $|b_i|< \delta $ for all $i$ from $1$ to $k$, then $(a_1+b_1),$$(a_2+b_2),$$... (a_k+b_k)$ are also linearly independent.

I believe it relates to the previously proven result that the set of full rank matrices in $\mathbb{R}^{k \times n}$ form an open subset of $\mathbb{R}^{k \times n}$. However, I'm not entirely sure how to apply this proposition in practice.

I would be grateful for any guidance here.