When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?
Solution 1:
Ask a more experienced person. IMHO that's really the only option, and one of the reasons for this is that it is very important for a proof to communicate a result and its justification to another person. If the proof is good enough to convince yourself, that's a start, but the real test is whether you can express it in such a way as to convince someone else.
And BTW... the same applies if the textbook does have a solutions manual. Your proof is inevitably going to be different from the one in the book, and it takes a lot of experience and mathematical understanding to decide whether the differences are important or not.
Solution 2:
1) This is your teacher's job to check your proofs. He is experienced and trained to read proofs and determine what is acceptable and suitable for your level (is it safe to ignore some minor flaw? How detailed a computation should be?...) Don't hesitate to ask advice on proof writing, including on non required work and self-study.
2) Do a step-by-step verification: are all formulas correct? Are all equivalences really equivalences? In particular, don't be lazy in that step: really check that the equivalences you used are not implications. Also, check carefully all the conditions before applying a theorem: the Alternating Series Test requires a decreasing sequence? check it! Including obvious conditions: if it is obvious, write it in one or two lines. As a teacher, it always upset me when students complain about their grade for an obvious fact they did not bother to state.
3) In order to help the verification in step 2, take some writing habits: for example, introduce the notations for the objects you're working on, don't write equivalences, only implications (i.e. to prove $P \Leftrightarrow Q$, prove $P\Rightarrow Q$ and then $Q\Rightarrow P$); prove equality of sets by double inclusion; a proposition starts by "for any $x$ in $A$,...", then start writing your proof by "Let's $x$ in $A$. Then...". Also, even if math is not high level litterature, good writing skills are absolutely necessary: especially, logic connectors like "if", "then", "so", "therefore", "but", "since",... have a precise meaning. Be sure you use them properly, since IMHO they greatly help to structure a proof and keep things clear.
4) Finally, math is not divided into two steps, one when you receive a lecture with definitions and proofs from the teacher like a sacred text, and a step where you do homeworks and try to copy the master. To be critical on your own work, you have to be critical on others'work. Efforce yourself to be question the professor's proofs: why did he introduce this? Can we shorten the proof like that? He used a non-intuitive trick in the proof; can we do it without the trick?
Solution 3:
In many proofs, the hard thing is to find the way which path to follow; once that is done they are easy. Faced with a problem, most of the time my result is: "I have no clue how to solve this", "I have a start but I'm stuck at some point", or "I have a proof which is correct unless I made a stupid mistake".
If you have no confidence that your proof is correct apart from possible mistakes, then you likely don't have a proof. If you say "A => B because I say so" in your proof, especially if you say "A => B must be true because otherwise my proof doesn't work", then most likely you don't have a proof. If that isn't the case, then most likely it's just a matter of checking your proof for mistakes.
Anyway, there are many exercises, too many to do them all. To learn, it isn't necessary to do all the proofs to the last excruciating detail; it's enough to get to the point where you can say "if I spent another hour or two then my proof would be faultless". Training your brain to get the right ideas so you can find proofs is the important thing. For new results, you want faultless proofs (and to avoid failing a test :-) For exercises, someone has written a faultless proof at some time.