Concept guide
Bisection: four bracket updates, the invariant and the separate proof obligation for root existence
For f(x)=x²−2, start from [1,2] with the supplied endpoint signs f(1)<0 and f(2)>0, run four bisection steps with the supplied midpoint signs and trace the brackets; prove that 'opposite endpoint signs and a<b' is an invariant, that the width after n steps is 1/2^n, and that, in an extended run with every needed sign available and effective updates, a width-based stopping rule terminates after finitely many steps; then show that the bracket data alone do not prove a root exists, and separately cite the intermediate value theorem to conclude, for the continuous f, that the final bracket contains a root.