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generalization error bounds

Hi!

I am having difficulty grasping the steps taken in 'a little thought' to obtain expression seen here.
q6.JPG

I am also not sure what 'by union bound' means, or specifically which theorem/lemma is used to claim it is at most K times, although intuitively it makes sense
q7.JPG

union typically means A or B but not A and B but in this case we are asking at least one event so not really the same as union

thanks in advance

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