# Sequences and limits

1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used.

Note: Here's Theorem 3.2.3

(a) Let X = (xn) and Y = (yn) be sequences of real numbers that converge to x and y respectively, and let c be an element R. Then the sequences X+Y, X-Y, X∙Y, and cX converge to x+y, x-y, xy, and cx, respectively.

(b) If X = (xn) converges to x and Z = (zn) is a sequence of nonzero real numbers that converges to z and if z ≠ 0, then the quotient sequence X/Z converges to x/z.

2) Let x1 > 1 and xn+1 : = 2 - 1/xn (for n an element of N). Show that (xn) is bounded and monotone. Find the limit.

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#### Solution Preview

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1) If (bn) is a bounded sequence and lim(an) = 0, show that the lim(anbn) = 0. Explain why Theorem 3.2.3 cannot be used.

Proof. Since (bn) is a bounded sequence, there exists M>o such that

|bn|<M for all n

Since lim(an) = 0, by definition , we have

given , there exists N>0 such that for

So,

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#### Solution Summary

There are several examples of working with sequences and limits to complete broofs.