proof that $\ln(x)$ is continuous at $x=1$ using $\epsilon-\delta$

Question

can anyone tell me, if this proof is correct:

for every $\epsilon>0$ there is a $\delta>0$ such that whenever $|x-1| < \delta$, then $|f(x)-f(1)| < \epsilon$. $|\ln(x)-\ln(1)|=|\ln(x)| < |\ln(0.5)| < \epsilon$ So: $δ=0.5$, $\epsilon > |\ln(0.5)|$