A common mistake students make when writing proofs is to start with the conclusion (the end) and end with the hypothesis (aka the beginning or the assumptions given in the question). While starting from the conclusion is a common technique to derive a proof in rough working, a formal proof should generally be written by starting with the assumptions and ending with the conclusion.
Consider the following problem and choose all the valid solutions from the options given below.
Prove that if and then for .

Question

A common mistake students make when writing proofs is to start with the conclusion (the end) and end with the hypothesis (aka the beginning or the assumptions given in the question). While starting from the conclusion is a common technique to derive a proof in rough working, a formal proof should generally be written by starting with the assumptions and ending with the conclusion. 
 Consider the following problem and choose all the valid solutions from the options given below. 
 Prove that if    and    then    for   .

Answer

Since   , we have   .  We know that   , so   .  This implies that    or   . But   , so we must have   , i.e.,   . 
 Therefore we have proved that if    and    then   .

Answer

Since   , by dividing both sides of    by    we obtain   . Therefore we have proved that if    and    then   .

Answer

Assume that   . By multiplying both sides by   , we obtain   , which is true. Therefore we conclude that   .

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A common mistake students make when writing proofs is to start with the conclusi...

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