Let x be an irrational number with convergents Pnf qn. For every n :'.::: 0, verify the following:1 I Pn I 1(a) < x - - < .2qnqn+l qn qnqn+l(b) The convergents are successively closer to x in the sense that[Hint: Rewrite the relationI Pn I I Pn-1 x-- < x---Iqn qn-1Xn+lPn + Pn-1x= ------Xn+lqn + qn-1as Xn+1(xqn - Pn) = -qn-1(x - Pn-ifqn-1).]

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