can someone explain this part "|x - z| + |z - y|" is

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>>7841324
pic related the middle of the inequality

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>>7841328

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>>7841339

>>7841324
Triangular inequality.

>>7841347
so the middle here >>7841345 of the inequality is not coming from the definitions here >>7841339

i thought it would be like this |a - b| <= |z - a| + |z - b| < |a - b|/2 but this is wrong

can you please explain

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really i only just don't understand this exact part

>>7841391
did you try drawing the thing? goddamit

P is the set of points that are at a distance less than epsilon from x.
so z in P means |z-x| < epsilon

same for Q and y. z in Q means |z-y| < epsilon

>>7841398
the way i am understanding it now is that: if c is in U intersection V then by the definitions in my book i get |a - c| < epsilon where epsilon > 0 so |a - c| = 0 by another definition in my book so a = c.

this means that a belongs to the neighborhood of a

then . . . c belongs to the neighborhood of b also

what does the contradiction mean exactly

>>7841398
lol sorry |a - b| <= |a - b|/2

thanks for your help

>>7841535
i meant |a - b| < |a - b|/2