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?