Introduction to Real Analysis 3rd Edition solutions

Introduction to Real Analysis 3
If a, b e JRandb =F 0, show that: (a) lal = N, (b) la/bl = lal/lbl.

Introduction to Real Analysis 3
If a, be JR,showthatla+ bl = lal+ Iblif andonlyif ab :::O.

Introduction to Real Analysis 3
If x, y, z e JRandx ~ z, showthatx ~ y ~ zifand onlyif Ix - yl + Iy- zl= Ix- zl.Interpret this geometrically.

Introduction to Real Analysis 3
Showthat Ix - aI < Eif andonlyif a - E< X < a + E.

Introduction to Real Analysis 3
If a < x < b and a < y < b, show that Ix - yl < b - a. Interprethisgeometrically.

Introduction to Real Analysis 3
Find all x e JRthat satisfy the followinginequalities: (a) 14x- 51~ 13, (b) Ix2- 11~

Introduction to Real Analysis 3
Find all x e JRthat satisfythe equationIx+ 11+ Ix - 21 =

Introduction to Real Analysis 3
Findallx e JRthatsatisfythefollowinginequalities. . (a) Ix - 11> Ix+ 11, (b) Ixl+ Ix+ 11<

Introduction to Real Analysis 3
Sketch the graph of the equation y = lxi-Ix - 1

Introduction to Real Analysis 3
Find all x e JRthat satisfy the inequality 4 < Ix + 21+ Ix - 11 <

Introduction to Real Analysis 3
Find all x e JRthat satisfy both 12x- 31 < 5 and Ix + 11> 2 simultaneously. 1

Introduction to Real Analysis 3
Determine and sketch the set of pairs (x, y) in JRx JRthat satisfy: (a) Ixl= Iyl, (b) Ixl+ Iyl= 1, (c) Ixyl = 2, (d) Ixl- Iyl=

Introduction to Real Analysis 3
Determine and sketch the set of pairs (x, y) in JRx JRthat satisfy: (a) Ixl ~ Iyl, (b) Ixl + Iyl ~ 1, (c) Ixyl ~ 2, (d) Ixl - Iyl:::

Introduction to Real Analysis 3
Let E > 0 and [, > 0, and a e R Show that VE(a) n V.s(a) and V/a) U V.s(a) are y-neighborhoods of a for appropriate values of y. 1

Introduction to Real Analysis 3
Show that if a, b e JR,and a =Fb, then there exist E-neighborhoods U of a and V of b such that U n V =

Introduction to Real Analysis 3
Showthat if a, b e JRthen (a) max{a, b} = 4(a + b + la - bl) and minta, b} = 4(a + b -Ia - bl). (b) minta, b, e} = min{min{a, b}, e}. 1

Introduction to Real Analysis 3
Showthatif a, b, e e JR,thenthe"middlenumber"is mid{a,b, e}=min{max{a, b}, max{b, e}, max{e,an.