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Since the right hand side approaches zero for large n, this means that for any $\alpha>0$ there is a number N such that the inequality is true for any n > N.2-Now for n>1 :
${\alpha}\ln(n)> \ln(\ln(n)) \Rightarrow \,\, \alpha> \frac{\ln(\ln(n))}{\ln(n) } $
since $\alpha $ is an independent variable of n I can choose it as small as possible so thatSince the right hand side approaches zero for large n, this means that for any $\alpha>0$ there is a number N such that the inequality is true for any n > N.
Your argument is flawless.since $\alpha $ is an independent variable of n I can choose it as small as possible so that
it becomes lesser than the right-hand side .
Can you give a counter example for $\alpha$ and n that disproves my argument ?