Prove theorem that says "Convergence in probability implies convergence in distribution" for the case where Xn and X are continuous random variables.
a) Given t and e, show that P(X <= t - e) <= P( Xn <= t + P(|Xn - X| >= e). This gives a lower bound on P(Xn <= t)
b) Use similar strategy to get an upper bound on P(Xn <= t)
c) By pinching, deduce that P(Xn <= t) -> P(X <= t)