Showing that the intersection of countable events equals 1? How would I show that if $P(A_{i}) = 1$  for all $i \geq 1$, then $P\displaystyle\left(\bigcap_{i=1}^{\infty }A_{i}\right)=1$?
I am rather stuck on how to prove this out.  I could rewrite $\bigcap_{i=1}^{\infty }A_{i}$ as an intersection of the complements of $A_{i}$, and then try to show that the probability equals zero?   Because if the probability of the intersections of the complements is zero, then I can conclude that it's 1 for the unions. 
Any suggestions on the first steps, or a way to approach this would be great!

You're basically already there, now use countable sub-additivity and squeeze the value out:

$$1\ge P\left(\bigcap_{i=1}^\infty A_i\right)^c = 1 - P\left(\bigcup_{i=1}^\infty A_i^c\right)\ge 1-\sum_{i=1}^\infty P(A_i^c)=1$$

