If you use a concrete-security definition of security for a PRG, then this statement is true. The proof is a good exercise. If you know enough to pose the problem and to understand the definition of security for a PRG, you should be able to find the reduction proof without difficulty. Start by tracing out what the definition is saying.
A general comment on your analysis: Your analysis makes it sound like you are trying to guess at the answer. Guesses are good, but then you should follow them up with proof. In cryptography, our intuition can often be wrong. Therefore, the accepted standard is a proof. In particular, if you guess that the statement is true, your next step should be to try to prove it; if you guess that the statement is false, your next step should be to find an attack that disproves it. Give it a try! This is a fun problem, and not too hard.
P.S. This looks very much like a standard homework question. I don't know whether it is or not, so I'll just say this. If you are using this web site to answer homework questions, you are not only cheating your fellow students, you are also cheating yourself of the chance to learn the material. To learn cryptography, you must grapple with the problems and try to solve them on your own. You'll only learn through undergoing the process yourself; you won't learn nearly as much by reading other people's solutions. Think of it like learning to ride a bicycle: you can't learn to ride a bicycle by watching someone else ride it. Instead, you have to get on the bicycle yourself, accept that you'll fall or stumble a few times, and work through it.