By Jiri Lebl

A primary direction in mathematical research. Covers the genuine quantity procedure, sequences and sequence, non-stop features, the by-product, the Riemann fundamental, sequences of services, and metric areas. initially built to coach Math 444 at collage of Illinois at Urbana-Champaign and later stronger for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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**Extra resources for Basic Analysis: Introduction to Real Analysis**

**Sample text**

I) Let {xn } be a convergent sequence such that xn ≥ 0, then lim xn ≥ 0. n→∞ ii) Let a, b ∈ R and let {xn } be a convergent sequence such that a ≤ xn ≤ b, for all n ∈ N. Then a ≤ lim xn ≤ b. 3 we cannot simply replace all the non-strict inequalities with strict inequalities. For example, let xn := −1/n and yn := 1/n. Then xn < yn , xn < 0, and yn > 0 for all n. However, these inequalities are not preserved by the limit operation as we have lim xn = lim yn = 0. The moral of this example is that strict inequalities may become non-strict inequalities when limits are applied.

4. Let x1 , x2 , . . , xn ∈ R. Then |x1 + x2 + · · · + xn | ≤ |x1 | + |x2 | + · · · + |xn | . Proof. We proceed by induction. The conclusion holds trivially for n = 1, and for n = 2 it is the standard triangle inequality. Suppose that the corollary holds for n. Take n + 1 numbers x1 , x2 , . . , xn+1 and first use the standard triangle inequality, then the induction hypothesis |x1 + x2 + · · · + xn + xn+1 | ≤ |x1 + x2 + · · · + xn | + |xn+1 | ≤ |x1 | + |x2 | + · · · + |xn | + |xn+1 |. Let us see an example of the use of the triangle inequality.

We leave this to the reader as a challenging exercise. We may also want to take the limit past the absolute value sign. 7. If {xn } is a convergent sequence, then {|xn |} is convergent and lim |xn | = lim xn . n→∞ n→∞ Proof. We simply note the reverse triangle inequality |xn | − |x| ≤ |xn − x| . Hence if |xn − x| can be made arbitrarily small, so can |xn | − |x| . Details are left to the reader. 3 Recursively defined sequences Once we know we can interchange limits and algebraic operations, we will actually be able to easily compute the limits for a large class of sequences.