Your admonition may be sound for other reasons, but your

argument is fallacious. Your example is a martingale, meaning the average gain is zero every time. The distribution of the gain is skewed with lopsidedly large gains concentrating in a small case, and money-losing at the most likely case in terms of total number of wins and looses. But that is not the determining factor for investment.

Here is a counterexample to your argument. Suppose the stock has 99.9% of the chance to grow 1000 fold and 0.1% of the chance to lose its value completely. So on average, the stock is hugely profitable.  Yet, your argument says you will not invest since almost surely you will lose all your money as n approaches infinity. But that only means either you are unbelievably risk-averse (cannot sustain loss or failure at any rate, which excludes all activities or careers your son could ever choose from) or using the worst strategy. With only a minuscule risk-tolerance, the optimal strategy is to invest almost all (in fact 99.999...%) of your wealth every time. Even if you have very high risk aversion, you should still invest no less than 90% of all your wealth in this stock every time. Even a constant amount, rather than fraction, of your wealth invested in this stock every time period will net you great gains. 

In general, the catch is to invest a fraction instead of all of your wealth. That fraction depends on the state probabilities and payoffs in each state. Depending on one's risk-aversion or utility function, that optimal fraction is called the (fractional) Kelly criterion.