The sample mean of a Normal RV does tend towards the population expected value, sure.

The GP comment is talking about a Cauchy RV, which has heavy tails. So it has enough probability mass at large values that the expected value is infinite. Discarding constant scale factors, in this case:

  E[X] = Int{0..infty} x * p(x) dx 
       = Int{0..infty} x * (1/x^2) dx 
       = Int{0..infty} (1/x) dx 
       = +infty

So, the sample mean of Cauchy random variates will not converge to any real number.