This question is about applying probabilistic bound (Hoeffding inequality) on proof.
The following proof is from Stanford cs229 Machine Learning course's problem set #2-5, proof for uniform convergence on erroneous distribution. http://cs229.stanford.edu/materials/ps2_key.pdf
The author used Hoeffding inequality (6) for step (9)->(10) and setp (11)->(12). If Hoeffding inequality is applied twice, Should the probability for equation (12)~(15) be not $1-\delta$, rather $(1-\delta)^2 $?
Any comments would be deeply appreciated. Thanks.