Random Variable Qusetion - A Level Further Statistics

Hi I'm working on a question:
"The random variable X follows a uniform distribution with the sample space {1/n,2/n,3/n,...,(n-1)/n,1}. Show that, as n tends towards infinity, E(X) -> 1/2 and Var(X) -> 1/12."
I have written the sample space in a table and tried to expand E(X) to 1(1/n)+2(2/n)+3(3/n)+...+(n/1)((n/1)/n)+n(1) = 1/2 and I think it has something to do with series summations but I'm not sure how to continue the question.
Any help would be appreciated, thanks.