What makes a sufficient proof? This question is related to the question posted here. Would a shorter proof to those in the answers, such as:

Take the subsequence $\{a_m\}$ of $\{a_n\}$ where $m > 0$. By induction on $m$ $$\forall{m}( a_m < a_{m+1})$$ Therefore $$ \forall{\epsilon} > 0 \ \exists{N} \ \forall{m_0,m_1} \ (m_0,m_1 \geq N \ \wedge \ m_0 \geq m_1 \ \ \Rightarrow \ \ m_0 - m_1 < \epsilon)$$ and $\{a_m\}$ is Cauchy. Hence it converges, so its parent sequence converges. Therefore its parent sequence is Cauchy.

be acceptable? 
Furthermore, what is the minimum amount of detail required in a proof? I am curious because I am switching into a maths program, however all my maths is self taught. I am not used to having my proofs reviews or having to prove to anyone but myself.
 A: This answer solely regards the latter portion of the question.
Whenever you write a proof, it is vitally important that you know your audience. This is key because a proof is simply an argument intended to convince the reader that the purported statement is true (in general, only automatic theorem provers actually show all the details).  So the question then becomes: "how much detail should I leave out?" This is precisely where your audience comes into play. For example, a professor of mathematics can, in general, make greater logical leaps than an undergraduate (for a variety of reasons such as familiarity with the material, etc). Thus the proofs presented to the undergraduate student would need to be more detailed than those presented to the other party.
However, there is one significant exception to the above rule: proofs written for coursework. In this case, the student is writing the proof for the professor to read, and so the professor could handle larger logical leaps. But, the student shouldn't write this way because the professor cannot tell if the student could actually fill in the details omitted. So, the student should write a proof as if he/she is communicating it to his/her classmates. Doing this should correspond to a level of detail where the audience (the professor) knows that the student has sufficient understanding of the material.
In summary, the level of detail required depends on the audience for whom you are writing. 
A: I can say the following about the required detail in proofs in math programs. Your teachers have to tell the difference between provers and posers. Therefore, your proofs should be quite detailed. You can skip anything that is explained in the lecture notes or in the books provided that you properly refer to them; besides that your can skip some high school stuff like standard limits, but not much else.
