  Let $W_{\lambda,b}(x)=\sum_{n=0}^\infty\lambda^n g(b^n x)$ where
$b\geqslant2$ is an integer and $g(u)=\cos(2\pi u)$ (classical Weierstrass
function). Building on work by Ledrappier (1992), Bar\'ansky, B\'ar\'any and
Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the
Hausdorff dimension of $W_{\lambda,b}$ equals $2+\frac{\log\lambda}{\log b}$
for all $\lambda\in(\lambda_b,1)$ with a suitable $\lambda_b<1$. This
reproduces results by Bar\'ansky, B\'ar\'any and Romanowska without using the
dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988),
which is replaced by a simple telescoping argument together with a recursive
multi-scale estimate.
