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319HW4Correction

# 319HW4Correction - MAT 319/320 Correction of HW4 Exercise 1...

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MAT 319/320 Correction of HW4 Exercise 1. Page 67, #6a. Proof. By the sum rule, the limit of (2+1/ n ) is equal to 2 . By the product rule, the limit of (2+1/ n ) 2 is 2.2 =4 . square Exercise 2. Page 67, #9. Proof. One has y n = n +1 - n = 1 n +1 + n (multiply the numerator and denominator by the conjugate quantity). But now one has 0 lessorequalslant y n lessorequalslant 1 n . Therefore if one proves that (1/ n ) converges to zero, the squeeze theorem implies that ( y n ) converges itself to zero. Fix any ε > 0 , then by the archimedean property there exists a natural number K such that K > ε 2 , but this implies that for any n greaterorequalslant K one has n > ε 2 , implying 1 n < ε , thus we proved that (1/ n ) converges to zero, and hence ( y n ) converges to zero. Now n y n = n n +1 + n = n n . 1 (1+1/ n ) +1 = 1 1+ 1+ 1 n . By the square root theorem 1+ 1 n radicalBig converges to 1 . By the quotient theorem (which applies because the limit of the denominator is nonzero), one knows that n .y n converges to 1/2 .

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319HW4Correction - MAT 319/320 Correction of HW4 Exercise 1...

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