These  Real Analysis appendices continue the decimal viewpoint of limits, continuity and convergence in chapter 14. and this further lesson

A. What's Next
B. Pigeon Hole Principle
B. Bolzano-Weierstrass
C1. Triangle Inequality
C2. Triangle Inequality
C. More T.Inequality
D. Sets & Sequences
D. Monotone Sequences
E. Limits,  Properties
E Limits & Error Control
F. Continuous Functions
F. Closed Range Thm
F. Intermediate Val. Thm
F. Compactness Thm
F. Equicontinuity Thm
F Extreme Value Thm
G. Rolle's Theorem etc
G. Mean Val. Thm.
G. Constant Difference Thm
G. Lipschitz Continuity I
PS: One Sided Range Theorems
G. Velocity Revisited
G. Sufficient Conditions
H. Riemann Sums Conv
H. Lipschitz Continuity II

Proofs of  one-sided theorems could be of interest in the study of 2D topology.

Appendix C
Triangle Inequalities

Assume x and y are real numbers. Then the first triangle inequality

|x+y| < |x|+|y|

holds with equality if and only if x and y have the same sign, or one is zero. Likewise, the second triangle inequality

|x+y| ³ max (|x|-|y|,|y|-|x|) = ê |x|-|y| ê

holds with equality if and only if x and y have opposite signs, or one is zero. The proofs of these inequalities are omitted.

If x = a+ib = (a,b) and y = c+id = (c,d) are complex numbers or points in the plane the first and second inequalities also hold as well. In this situation, the points (0,0), x = (a,b) and x+y = (a+c,b+d) form the vertices of triangle with sides of length |x| = Ö[(a²+b²)], |y| = Ö[(c²+d²)] and |x+y| = Ö[((a+c)²+(b+d)²)]. The first triangle inequality |x|+|y| ³ |x+y| implies or reflects the observation that the sum of the lengths of two sides of a triangle in the plane is greater than the length of the third. The second triangle inequality |x+y| ³ | |x|-|y|| indicates the length of the third side is greater than the difference of the lengths of the other two. See the following diagram.

The first triangle inequality indicates the length of the arrow x+y forming one side is less than (or =) sum the length of the other two sides. The second triangle inequality indicates the length of the arrow forming one side is also greater than (or =) the differences of the lengths of the other two sides.

Postscript: Outside of this site, the webpage Triangle Inequality offers a simple proof of the triangle inequality in which
<a,b> = a₁b₁+a₂b₂2+a₃b₃

denotes the inner product of points in space.

By mathematical induction, the first triangle inequality can be extended as follows. Triangle inequalities are of interest in estimating or bounding the size or magnitudes of real and complex numbers (error control say). Examples or applications will follow in the appendix Properties of Limits.

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