A geometrical inequality with a sphere

Question

A bee flied 4 meters (in total) and back to the original spot. Prove its path can be inscribed in a sphere with radius 1m.

I am bad at $3D$ geometry, so I tried to reduce it to a plane, but I still can't solve it.

Any help appreciated.

Answer 1

If you have a rope of length $L=4m$, tie it into a closed loop and tack it to the ground at one point, how much can you extend this rope loop at most?

The obvious answer is that you fly straight to some target destination and return straight back. This can reach a destination $2$m away. At the midpoint you can put the rope loop into a sphere of radius $1m$.

If one wants to prove this with rigour, I fear one has to formulate this as an instance of the general isoperimetric problem of the calculus of variations. (E.g. see here)

So how much rigour do you need?