Triangle Inequality |x +y| < |x| +|y| 

The following diagrams illustrate and imply the triangle inequality in for real numbers and for points in the plane.

Geometric View in one dimension

Every real number x and y can be identified with a displacement, arrow, vector or directed line segment.

The length of x + y that is |x+y| equals the sum of the lengths |x| and |y| when  x and y have the same direction.

The length of x + y that is |x+y| equals the difference of the lengths |x| and |y| when  |x| > |y|.

Therefore  |x| > |y| 

||x| - |y|| = |x| - |y| < |x +y| < |x| +|y| 

Likewise when |y| > |x| 

||x| - |y|| = |y| - |x| < |x +y| < |x| +|y| 

So in both cases 

||x| - |y|| < |x +y| < |x| +|y| 

Geometric View in Two Dimensions

Every point x and y in the plane can be identified with an displacement, arrow or vector.

When  |x| > |y| and |y| = r > 0, the length L =  |x+y| of of x+y  satisfies

  |x| - r < L < |x| + r

or equivalently

||x| - |y|| = |x| - |y| < L = |x +y| < |x| +|y| 

Likewise when |y| > |x| 

||x| - |y|| = |y| - |x| < |x +y| < |x| +|y| 

So in both cases 

||x| - |y|| < |x +y| < |x| +|y| 

Remark: The explanation or image in three dimensions is similar:

Remark:  When |x| > |y| >0 equality may hold in exactly one of the two inequalities

||x| - |y|| < |x +y|    and |x+y| < |x| +|y| 

only when x and y are collinear.

Remark: Algebraic proofs dependent on the Pythagorean distance formulas for the distance between points and for the lengths of vectors are available.  Students of pure mathematics will or should see them. 

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2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
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3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
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3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
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3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4.  Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5  Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere

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Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity

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