help convergence in distribution

If there is b1,b2, a sequence with non negative real numbers, and the sum of them is 1. Also P(X=i)=bi for all positive integer i. How do I make a sequence {Xn} of absol. cont. R.V's such that Xn converges in distribution to X.

What i have so far:
b1>0 b2>0 , ... , bi>0
P(X=1)=b1 , ... , P(X=i)=bi
so Xn has c.d.f
b1ϕ(x) + ... + biϕ(x)
(the mean of each phi above increases by 1 and starts at 1)
(theres also a subscript after the phi on the bottom that increases by 1 and starts at 1)

Does this make sense?