By Jiri Lebl

A primary path in mathematical research. Covers the genuine quantity approach, sequences and sequence, non-stop features, the by-product, the Riemann crucial, sequences of capabilities, and metric areas. initially constructed to educate Math 444 at collage of Illinois at Urbana-Champaign and later better for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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Let {xn } be a convergent sequence and let x := lim xn . First suppose that x = 0. Let ε > 0 be given. Then there is an M such that for all n ≥ M we have √ xn = |xn | < ε 2 , or in other words xn < ε. Hence √ √ √ xn − x = xn < ε. Now suppose that x > 0 (and hence √ x > 0). √ √ xn − x √ xn − x = √ xn + x 1 √ |xn − x| =√ xn + x 1 ≤ √ |xn − x| . x We leave the rest of the proof to the reader. 1/k A similar proof works the kth root. That is, we also obtain lim xn = (lim xn )1/k . We leave this to the reader as a challenging exercise.

X∈D x∈D b) Find a specific D, f , and g, such that f (x) ≤ g(x) for all x ∈ D, but sup f (x) > inf g(x). 7 without the assumption that the functions are bounded. Hint: You will need to use the extended real numbers. 4. 5–1 lecture (proof of uncountability of R can be optional) You have seen the notation for intervals before, but let us give a formal definition here. For a, b ∈ R such that a < b we define [a, b] := {x ∈ R : a ≤ x ≤ b}, (a, b) := {x ∈ R : a < x < b}, (a, b] := {x ∈ R : a < x ≤ b}, [a, b) := {x ∈ R : a ≤ x < b}.

N→∞ n→∞ For a bounded sequence, liminf and limsup always exist. It is possible to define liminf and limsup for unbounded sequences if we allow ∞ and −∞. It is not hard to generalize the following results to include unbounded sequences, however, we will restrict our attention to bounded ones. Let us see why {an } is a decreasing sequence. As an is the least upper bound for {xk : k ≥ n}, it is also an upper bound for the subset {xk : k ≥ (n + 1)}. Therefore an+1 , the least upper bound for {xk : k ≥ (n + 1)}, has to be less than or equal to an , that is, an ≥ an+1 .