I am still confused about what happened in 28:40. Why can we just simply add something to the left and then change the inequality the other way around?
Because that's the definition of the infimum: For all a in A, and for all epsilon > 0, a - epsilon < inf A =0, and epsilon >0, epsilon/2^k > 0. See "Infima and suprema of real numbers": en.wikipedia.org/wiki/Infimum_and_supremum Normally the inequality is strict but by taking epsilon arbitrarily close to zero the inequality becomes loose.
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I am still confused about what happened in 28:40. Why can we just simply add something to the left and then change the inequality the other way around?
Because that's the definition of the infimum:
For all a in A, and for all epsilon > 0, a - epsilon < inf A =0, and epsilon >0, epsilon/2^k > 0.
See "Infima and suprema of real numbers": en.wikipedia.org/wiki/Infimum_and_supremum
Normally the inequality is strict but by taking epsilon arbitrarily close to zero the inequality becomes loose.
lambda star is the infimum of the RHS. thats why adding a number greater than zero flips the inequality
If you add a positive number to the greatest lower bound of a set, that number is no longer a lower bound of the set.
Could this criteria lead from countable subadditivity to countable additivity?
disjoint unions for countable additivity
Isnt finite union concluded from the 2 sets union proof ? I guess u could prove it but shouldnt it be trivial
Why use inequality and add the epsilon if we will just equal it to zero . Then why not just use equality .