Problem 2(5pts)Consider the following four "proofs" that eis irrational."Proof" 1 differs from the lecture notes in that b has been replaced with b+2in most places."Proof" 2 differs from the lecture notes in that b has been replaced with b-2in most places."Proof" 3 has been rewritten to not use contradiction."Proof" 4 has used the phrase "without loss of generality" to greatly simplify the proof.For each "proof", determine if the change isan error or not and explain why or why not.Theorem. Euler's number, e,is irrational."Proof" 1. Suppose for contradiction that eis rational. Thus, there exist natural numbers a and bfor whiche=abApplying the definition ofe and multiplying by(b+2)!,we haveab(b+2)(b+2)!*e=(b+2)!*i=01i!=i=0(b+2)!i!=i=0b+2(b+2)!i!+i=b+3(b+2)!i!Since ab(b+2)!isan integer, and i=0b+2(b+2)!i!isan integer, we must have thati=b+3(b+2)!i!is also an integer. Note that for all ib+3,(b+2)!i!1(b+3)i-(b+2)Soby comparison with the geometric series we have01absoeab.