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Blast into Math! Analatic nummer theora: ants, ghosts and giants
The triangle inequality means that the distance between two points on the number line, a , and b, is less
than or equal to the distance from a to c plus the distance from c to b . This means that going from
a directly to b is either shorter or the same as going from a to c and then from c to b .
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Exercise: When is |a − b| = |a − c| + |c − b| ?
If we use the triangle inequality with c =0, then
|a − b|≤ |a − 0| + |0 −−b| = |a| + |b|.
Now we can use this with x n − X as “a ” and Y − y n as “b ”
|x n − X + y n − Y | = |x n − X − (Y − y n )|≤ |x n − X| + |Y − y n | = |x n − X| + |y n − Y |,
because the distance between y n and Y on the number line is
|y n − Y | = |Y − y n |.
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