Could someone please check my work and let me know if it's correct

Please state all ce'w'xs an: thea'er's that V3.4 'nl need Deii'iztc'i' 5 i i Let j' 17 - '3 we Ni be m mum-main\" (\\m it o' I' We (at that a real numboi I Is a limit of I 2!: , 1i '5: mrh U t'w'n' rvsis J ,\\ ii surh that firi I ' whenever I I' and ii 3* .' 4' Dime 'lhmiem 5 I 10 let I it \\ '4. and let i be an accumulation point oi 17 . Then the following are equwalent: tn' 1 does not have a limit at 1" iii) Thom exists a sequence is\") in I) with each 5,, : i' such that (3,.) converges to 1' , but (fis,,)) is not convergent in :4: . i. (u) \\ iii) If f does not have a limit at (' then there exists a sequence (3,.) in D with each 3.. r v such that (s..) converges to v . but (f(3,.)) is not convergent in 11%. Suppose f does not have a limit ate and there exists a sequence (3") in D with each an 76 r and (3,1) -> c. Then by definition 5.1.1. since f does not have a limit at c , 1 V5 > 0 36 = Z (choosing (Ste be infinitesimally small) such that El sn 6 D such that 1 0 e ,which means (f(s,.)) will never be within E of a limit, L , which means (f(sn)) is not convergent to L . Since (f(.sn)) is not convergent to L , the sequence (f(s")) is not convergent in R. Therefore, if f does not have a limit at c then there exists a sequence (87,) in D with each s" 76 c such that (3,.) converges too, but (f(sn)) is not convergent in IR. 2. (b) => (a)|f there exists a sequence (3,.) in D with each 3,. 75 a such that (3,.) converges to c , but (f(sn)) is not convergent in IR then 3' does not have a limit at c . Suppose there exists a sequence (an) in D with each an g c such that (31,) converges to c and (f(sn)) is not convergent in R. Now suppose by contradiction that f has a limit at c . Then by definition 5.1.1, for each 5 > 0 36 > Osuch that 3.9,, E D such that 0