Zerotoepsilon

Hello! Here I am going to discuss some relation between Cauchy Sequences and functions. The main motive is to know what kind of function(continuous/uniform) preserves Cauchy sequences and vice versa. ➤ Does a continuous function always preserve a Cauchy sequence? We know sequential def. of continuity, so if we take a Cauchy sequence x_n on a closed subset of   ℝ  then it will surely converge to some point x in the domain, hence f(x_n) will converge to f(x). So our main goal is to deal with subsets that are not closed( they must have a Cauchy sequence that doesn't converge). So one such set is (0,1). Now take f(x) = 1/x and consider x_n = 1/n be a Cauchy sequence. f(x_n) = n, which is not Cauchy hence Continuous function need not preserve Cauchy sequence. ➤  Does a uniformly continuous function always preserve a Cauchy sequence? Suppose f: 𝔻 →  ℝ is uniformly continuous.  By def. of U.C. function, given  ε   > 0 ∃  a  δ   > 0 such that |f(x)-f(y)| <  ε whenever |

Here is fully solved paper of  TIFR GS 2010  with justification of each and every step, hope you'll enjoy it. For any query feel free to write at  sushantkala786@gmail.com .                                              PART  -  'A' Q.1   Ans.          (c).  phi(60) = 16 Justification   -:     We know that in a cyclic group G of order n generated by a. The order of an element a^(k) is n/(n,k)=n iff (n,k)=1. So the number of generators of G are phi(n). Q.2     Ans.          (a). (a) False,  Consider the direct product Z_3*Z_3*Z_3. (b) True,   Any Abelian group of order 14 must-have elements a and b with respective order 2 and 7 ( Cauchy Theorem ). Then a*b generates the group. (c) True, Same logic as in option (b). (d) True, Same logic as in option (b), (c). Q.3    Ans.           (c). 6 Justification   -:     Calculating last digit is the same as 2^(80) mod (10). We know that 2^(5) =2 (mod 10). So 5 can be treated as 1 in power.  Since 80 =3*5^(2)+5,