Still struggling with proofs. How do you construct rigorous math proofs on your own? Also how do you verify? I am finishing up my first semester of undergraduate analysis and still am struggling with writing proofs. Even though I am getting an A, I just do not feel confident that I can clearly write proofs on my own. 
 A: It takes practice to get good at anything. You're just getting started. Visit the Professor during office hours; most faculty like questions from serious students such as yourself. Ask the faculty to recommend good books to help you with proofs.
A: As I said in my comment, without knowing where you get stuck it's hard to give good advice. I can, however, try to explain what I usually do when I try to prove something.
First, I try to "see through" the mathematical notation and understay the underlying actual problem. E.g, for geometric problems, the actualy problem could be something about the various ways you can or cannot place points on a plane, or maybe having lines intersecting a circle, or whatever. The important thing is that notation is important tool to communicate your ideas, but not always a good way to think about them.
Then I play with the problem. I usually start with simple cases where the theorem is obviously true. Then I move on to trying to construct a counter-example. In effect, I'm playing a game against the theorem - I try to be mean, to kind of "force" the theorem to be wrong, by looking for cases where it cannot possibly hold. If I suceed, I'm done - I have disproven the theorem.
If I don't suceed, I try to understand why the theorem always wins - try see why I cannot construct a counter-example, why there's always a "way out" for the theorem that makes it still true, no matter how hard I try. That's usually the hardest part, but also the one that's the most fun.
Finally, I try to formalize my thoughts. There's were notation gets back into the game. I take the usually quite informal reason I came up with in the previous step, and try to translate it into an actual proof, step by step.
A: Further to what's been said above, your textbooks are a good source of verifiable proofs.  If you stop every time you reach a proposition, Lemma or Theorem in the textbook and attempt to prove it yourself before reading on, you not only get practice at constructing proofs, but you have an "ideal" answer to look at afterwards.
You won't get all of them; sometimes a proof depends on a clever idea that is obvious when you see it but hard to think of on your own.  However, (good) textbooks tend to provide you with the concepts needed for each proof in a constructive manner so you should get enough of them to not lose confidence.
And don't think that the textbook gives the only proof either; there are often many ways to prove a fact (look at the variety of answers questions on this site get, for example).
