Question

Claim: {(-1)"} does not converge to any real number a. Proof: Assume that the sequence converges; that is, assume that there is an a E R such that lim,--.(-1)" = a. Then, using E = 1, from the definition of convergence, we know that there exists an no such that |(-1)" - al < 1 for all n > no. Thus, for any odd integer nno, we have |(-1)" - al = 1-1-a[< 1, and for any even integer n> no. 1(-1)" – al =1+1 -al < 1. The number a is within one unit of both - 1 and +1, a contradiction. 6. Using the example above as a guide, prove that the sequence sin any real number a. e { sin(97)} does not converge to

Answer 1

6) het it possible lim sin (A) = a (any real number a) n> an 'no no such then by definition of convergence, there Edirt that (na) - al<t for < 1 for all no no. if na 1,5,9,13 ~- that if na 3,7,11,15-- I sin ( sup>3> { -1 this for any ny, no of type 1,5,9,13 o ☆ C | sin (1937) -a) | 1-al< 1 for all nano. and for any none of type , 317, 11,25 | sin (1995)- a) = |--akt for all ny no -

which is Contradiction to and as à is within one unit of both - and H. Hence. servence {sin ht} does not converges to any real number ca.
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