TITLE: Graduate school self-doubt (currently an undergraduate)
QUESTION [25 upvotes]: I don't know if this is the right place for such a question, so please redirect me to a more appropriate place if necessary.
My main question is, if you go to a small school, are you doomed to fail right from the get-go? My secondary question is, what are my options then, if I am indeed doomed to "fail"? The following is more of a personal description, so obviously it doesn't apply to everyone, which would make the question "not general enough", but I suspect that there are many others caught in a similar situation.
I'm going to be a senior (fourth-year) in the fall (August) and am currently doing my "homework" on graduate school admissions. I go to a small liberal arts college you've probably never heard of. From what I've read and heard, I'm extremely discouraged and I'd say I'm borderline depressed about my future, but the truth is, I don't really want to do anything else other than learn more math (and get a PhD in the process if I'm capable of it).
I like to think that I've worked very hard (but everyone says the same thing). Classes I've taken include topology, complex analysis, algebra, real analysis, statistics, graph theory, number theory, and Galois theory. I've been told that none of my coursework means much to graduate schools because my school is relatively unknown. Furthermore, much as I can get some good recommendation letters, I've heard that since none of my letter writers is "relatively well-known" in their area, and everyone who applies to graduate schools gets glowing reviews anyway, I can't compete with other applicants at all.
I've gone through all of Herstein, a little more than half (i.e., as much as I could) of Rudin, most of Hoffman/Kunze, and the first half of Munkres. None of these texts are actually used at my school. In other words, I don't think I've given any less than I'm physically capable of in the last three years, and if that isn't enough, I really shudder to think what is.
I haven't taken the GRE yet but am currently studying for it. But it's no secret that the GRE is a relatively silly measure of grad school success. A bad score would condemn me to eternal hell, but a good score isn't a direct ticket to success.
I've heard the "anything can happen ... who's to say that you can't succeed ... people from small schools have gotten into big schools" argument, but in all realisticness, there's absolutely no reason to pick me over a Princeton graduate who has the exact same credentials, and there are tons of them. So what are my best options given this reality?
Any advice/encouragement/experiences would be very much appreciated. Thank you.

REPLY [2 votes]: Haha, you're worried waaay too much! If you're studying out of those books and have a good solid foundation on what you've learned, then you should be fine. However, the probability of you being accepted, and as a consequence passing, any PhD program in Mathematics is up to you. Allow me to explain what I mean (though this was posted 3 years ago, it may be beneficial to anyone who has the same question now).
You're an undergraduate now. What do you have to show for the fact that you are actually a mathematician and that you can not only pass graduate course classes and advance the field? Here are just some example questions you may want to consider:

How are your grades in your courses? Is your GPA high?
How much did you retain from your courses? For example, if I asked you what the definition of a vector space is, could you list out the criterion for me? 
What research have you done in the field? This is often a really big determining factor on acceptance in graduate school, especially if you've done research and made rediscoveries of your own. 
Was your research useful? If you're working in a particular field, did your rediscovery contribute to the field itself? 
Could you actually handle the course-load? This isn't just a factor of timing or difficulty. Recruiters will base this off of your GPA, and your GRE, since those are the most official ways to test your knowledge and ingenuity. However, you will be asked what you're going to shoot for in the future. Make sure you do your homework on the school, what kind of mathematics the school mostly centers around, and the results they've recently ascertained through research. Speaking of which,
Are you up-to-date on the most recent mathematical discoveries and unsolved mathematical problems? This isn't necessarily a big issue, but it may help out with decision making, along with some of the factors above.

Most students do not have all of this criteria, and make it into graduate school in mathematics, but the biggest things are your thesis and your GPA. The books you're using is fine. If it helps, talk to your research advisor and get his/her advice on it as well. Hope this helps!