In a convex lens of focal length F, the minimum distance between an object and its real image must be:

Let image distance be v, object distance be -u and focal length be f.
From lens formula, 1/v - 1/u = 1/f
Let distance between object and image be s, s = u + v
1/v + 1/u = 1/f
Substituting v = s - u,
1/(s-u) + 1/u = 1/f
s/(u(s-u)) = 1/f
su = u(s-u)/f
s = u(s-u)/f
fs = us - u²
u² - us + fs = 0
For real values of u, the discriminant must be ≥ 0
s² - 4fs ≥ 0
s² ≥ 4fs
s ≥ 4f
For point of minima, ds/du = 0
u²/f(u-f)² = 0
u = 2f
s = (2f)²/(2f - f) = 4f