- Find lim
_{n→∞}. - Find lim
_{n→∞}. - Find lim
_{n→∞}. - Find lim
_{n→∞}− n. Hint: Multiply and divide by+ n. - Find lim
_{n→∞}∑_{k=1}^{n}. - Suppose is a sequence of complex numbers which converges to the complex number x + iy. Show this happens if and only if x
_{n}→ x and y_{n}→ y. - For < 1, find lim
_{n→∞}∑_{k=0}^{n}r^{k}. Hint: First show ∑_{k=0}^{n}r^{k}=−. Then recall Theorem 4.4.9. - Using the binomial theorem prove that for all n ∈ ℕ,
^{n}≤^{n+1}. Hint: Show first that=. By the binomial theorem,Now consider the term

and note that a similar term occurs in the binomial expansion for^{n+1}except you replace n with n + 1 whereever this occurs. Argue the term got bigger and then note that in the binomial expansion for^{n+1}, there are more terms. - Prove by induction that for all k ≥ 4, 2
^{k}≤ k! - Use the Problems 21 and 8 to verify for all n ∈ ℕ,
^{n}≤ 3. - Prove lim
_{n→∞}^{n}exists and equals a number less than 3. - Using Problem 10, prove n
^{n+1}≥^{n}for all integers, n ≥ 3. - Find lim
_{n→∞}nsinn if it exists. If it does not exist, explain why it does not. - Recall the axiom of completeness states that a set which is bounded above has a least
upper bound and a set which is bounded below has a greatest lower bound. Show that
a monotone decreasing sequence which is bounded below converges to its
greatest lower bound. Hint: Let a denote the greatest lower bound and recall
that because of this, it follows that for all ε > 0 there exist points of in.
- Let A
_{n}= ∑_{k=2}^{n}for n ≥ 2. Show lim_{n→∞}A_{n}exists and find the limit. Hint: Show there exists an upper bound to the A_{n}as follows. - Let H
_{n}= ∑_{k=1}^{n}for n ≥ 2. Show lim_{n→∞}H_{n}exists. Hint: Use the above problem to obtain the existence of an upper bound. - Let I
_{n}=and let J_{n}=. The intervals, I_{n}and J_{n}are open intervals of length 2∕n. Find ∩_{n=1}^{∞}I_{n}and ∩_{n=1}^{∞}J_{n}. Repeat the same problem for I_{n}= (−1∕n,1∕n] and J_{n}= [0,2∕n). - Show the set of real numbers is not countable. That is, show that there can be no mapping from ℕ onto. Hint: Show that every sequence, the terms consisting only of 0 or 1 determines a unique point of. Call this map γ. Show it is onto. Also show that there is a map fromonto S, the set of sequences of zeros and ones. This will involve the nested interval lemma. Thus there is a one to one and onto map α from S toby Corollary 3.2.5. Next show that there is a one to one and onto map from this set of sequences and P. Consider
Now suppose that f : ℕ →

is onto. Then θ ∘ α^{−1}∘ f is onto P. Recall that there is no map from a set to its power set. Review why this is. - Show that if I and J are any two closed intervals, then there is a one to one and onto map from I to J. Thus from the above problem, no closed interval, however short can be countable.

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