By Mangatiana A. Robdera

This textual content introduces to undergraduates the extra summary strategies of complicated calculus, smoothing the transition from typical calculus to the extra rigorous process of evidence writing and a deeper realizing of mathematical research. the 1st half bargains with the elemental beginning of study at the genuine line; the second one half reports extra summary notions in mathematical research. every one subject encompasses a short advent and designated examples.

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**Example text**

G(y). (1) Let a E IR. Evaluate 9 (2a) , 9 (3a) , and more generally 9 (na) for all n E N. (2) Compare 9 (-x) and 9 (x). Write 9 (rna) in terms of 9 (a) and m, where m E Z. (3) Write 9 (ra) in terms of 9 (a) and r, where r E Q. Let f : X -+ Y be a function. Show that c (1) if A (2) if BeY, then (3) if BeY, then c f- 1 (f (A)); f (1-1 (B)) c B; f- 1 (Y \ B) = X \ f- 1 (B). X, then A Let f : X -+ Y be a function. Let {Ai: i E I} and {Bi : i E I} respectively be a family of subsets of X and Y. Show that (1) f (UiEI Ai) = UiEI f (Ai); 1.

Be a sequence of closed bounded intervals such that Suppose that lim (b n element. - an) = O. Show that the In have exactly one common 47 2. 3). 20, a = lim an exists as a real number. Since for each nand each p an :::; an+p :::; bn+p :::; bn, n:=l we have a :::; bp for every p. Therefore a E Ip for all p. Hence In 3 a. Suppose that b E In· Then b E [an, bnl for each n. Thus 0 :::; b - an :::; bn - an. It follows from lim (b n - an) = 0 that b = lim an = a. This completes 0 our proof. 44 more precise.

2 N N+1 N+2··· .... N+p Bounded increasing sequence Solution Let (an) be a convergent sequence and let lim an = a. For each n, we have Consider lanl < lal E = 1. There is N in N such that Ian - al all n > N. It follows that if + 1, for < 1 for n > N. Therefore then lanl ::; M for all n E N. o Note Bounded sequences mayor may not converge. The sequence (( -It)nEN is an example of a sequence which is bounded but not convergent. So boundedness is not appropriate for testing the convergence of sequences.