#### MathematicalPhysicist

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Q is the rationals set.

can be covered by open intervals I_k (k is natural number) which the total sum of their lengths is smaller than 1/100.

2. an integer number is called simple if you can write it by the letters ),(,1,2,+,* (* is multiplication) when we can use the letters at most 10 times.

i need to show that there exists a natural number N,

1<=N<=10000000000 which isnt simple.

for the first question im not sure,

i need to show that A is a subset of the union of open intervals which their total length is smaller than 1/100, obviously we can dissect A into,

(0,1/1000),(1/1000,2/1000),....(999/1000,1) but then we don't have the end points included.

perhaps the end points should be irrational, cause they anyway cannot be attained in the set A?

for the second question i dont have a clue, i thought perhaps give an ad absurdum proof, suppose every natural number in the interval [1,10000000000] is simple then the number of those numbers is 10000000000, and we have used at most 100000000000,

if we show that the cardinality of the set of letters used is smaller than the cardinality of the set of simple numbers in this interval than we have a contradiction, but im finding it hard to find a one to one function between sets, i dont know how to even to define one.

any help?