Harper’s operator is defined on ${\ell }^2\left(Z\right)$ by $$H_{\theta }\xi \left(n\right)=\xi (n+1)+\xi (n-1)+2cosn\theta \xi \left(n\right),$$ where $\theta \in [0,\pi ]$. We show that the norm of $\parallel H_{\theta }\parallel$ is less than or equal to $2\sqrt{2}$ for $\pi /2\le \theta \le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.