5. Let z be an irrational number. Prove that there exists a sequence (rn) of rational...

   

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5. Let z be an irrational number. Prove that there exists a sequence (rn) of rational numbers which converge to z. 6. Let (an) and (bn) be sequences with lim an = l and lim bn l. Let (cn) be the sequence defined by the = n→∞0 n-∞ formula Cn = an+1 bn if n is odd if n is even. In other words, (Cn) is the sequence (a1, b1, a2, b2, a3,b3,...). Prove that lim cn = l. n10 5. Let z be an irrational number. Prove that there exists a sequence (rn) of rational numbers which converge to z. 6. Let (an) and (bn) be sequences with lim an = l and lim bn l. Let (cn) be the sequence defined by the = n→∞0 n-∞ formula Cn = an+1 bn if n is odd if n is even. In other words, (Cn) is the sequence (a1, b1, a2, b2, a3,b3,...). Prove that lim cn = l. n10