Let $x$ be an irrational number and the rational number $/$ satisfy the inequality: $$ \bigg|−\frac{a}{b}\bigg|<\frac{1}{2b^2} $$ Then $/$ is a convergent of $x$
A proof by contradiction is given in Condition for Rational to be a Convergent.
But another proof is given in my reference for the case when $x$ is rational, Page 637,638, Appendix 4 : Number Theory, Quantum Computation and Quantum Information by Nielsen and Chuang, an equal sign is present in the equality, i.e., $$ \bigg|−\frac{a}{b}\bigg|\leq\frac{1}{2b^2} $$ Going through both the proofs I think equality is valid in the expression. I'd appreciate any help to validate it ?
Thanks @JohnOmielan for the hint.
Can I extend the proof by contradiction to the case when $x$ is rational and add an equal sign to the condition ?