Tweaking Preserves Linear Independence

Suppose $v_1,v_2,\ldots,v_n$ are $n\ge1$ linearly independent vectors over, say, the rationals, reals or $p$-adic numbers (in any case, denote the field by $F$). Suppose we perturb the vectors a bit to get $v'_1,v'_2,\ldots,v'_n$ (with $v'_i\approx v_i$, in the sense that for some small $\epsilon>0$ we have $\lVert v'_i - v_i\rVert < \epsilon$ for all $i$).

If the perturbation is small enough (i.e. for sufficiently small $\epsilon>0$), then $v'_1,\ldots,v'_n$ will still be linearly independent over $F$. Why?

One way to think about this is that we are going from the qualitative fact that "linear independence is equivalent to nonzero determinant" to a (more) quantitative fact about tweaking/perturbation, using the observation that the determinant (a polynomial in matrix entries) behaves "as expected" under small changes in inputs (formally, due to niceness (continuity) of polynomials).

(It is also possible to prove the above result without using determinants: try it yourself!)

For an example of an application in number theory (!), search "earthquake" in Eva Belmont's notes for MIT's 18.785 Fall 2014, taught by Prof. Bjorn Poonen.).