This is a long answer, give reasoning/calculations.

Show that with the number of observations n in a multiple regression being held constant, an increase in the number of dimensions/predictors p results a lower F-statistic for the null hypothesis that all regression coefficients are 0. What does this imply about regression results for high-dimensional data with small sample sizes?

Give a detailed answer with complete explaination of the argument and show proof. Mention each step.