Accept $ M_n $ is a $ n times n $ Matrix with independent entries $$ M_n (i, j) $$ Be $ 0.1 $ Bernoulli random variables, each with $$ mathbb {P} (M_n (i, j) = 1) = dfrac { ln n} {n} $$

How do we show that as $ n to infty $the probability that at least one column is zero is delimited $ 0 $ (ie with a positive probability)?

We can use the inclusion and exclusion principle to obtain a series and use mathematica as the first term in the series $$ n (1- dfrac { ln n} {n}) ^ n to 1 $$ We can probably use the alternate series test to draw the conclusion. However, it is difficult for me to calculate the above limit (and the limit of the other terms in the series) by hand.

Every help is appreciated.