We say that an infinite sequence a0,a1,a2,a3,… of realnumbers has the limit L if for every strictly positive number ?,there is a natural number n such that all the elementsan,an+1,an+2,… are within distance ? of the value L. In this case,we write lim a = L.

1. Express the condition that lim a = L as aformula of predicate logic. Your formula may use typicalmathematical functions like + and absolute value and mathematicalrelations like > or ? . Check that your formula has exactly twofree variables: a and L.
   

2. For any two infinite sequences of real numbers a =a0,a1,a2,… and b = b0,b1,b2,… we define their sum (written a+b) tobe an infinite sequence c0,c1,c2,c3,… such that ci = ai + bi forevery i?0. Prove (in complete sentences of “mathematical English”)that if we have two infinite sequences with lim a = L andlim b = M then lim (a+b) =(L+M).
   

3. Identify 5 places in your Part B proof where you took astep corresponding to a natural deduction rule, explicitly orimplicitly. Each place should correspond to a different naturaldeduction rule.
   

4. Your proof almost certainly used well-known mathematical“facts” about natural numbers or real numbers. These “facts” arejust small theorems that are so obvious or well-known that they canbe used in proofs without comment. Give predicate logic formulas tospecify three theorems you used. (no proofs for these, but theyshould all be true formulas about numbers, with no freevariables.)