How do we know F(x,y)=c level sets constrained by function G(x,y) = z will intersect when finding Lagrange Multipliers?

Question

I am struggling with Lagrange Multipliers proof. I am OK once I accept the fact that the constraint overlaps somewhere. Usually the proofs will say f(x,y)= c and g(x,y) = c. How do we know the level sets overlap somewhere ? Can someone explain? Thank you. P.S. once the overlap is established the rest of the proof almost seems a forgone conclusion if the gradient is perpendicular to the level set at that point.