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mathscott123
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Homework Statement
Suppose that a mathematically inclined child plays with a basket containing an infinite subset of integers (with some repetitions). If an integer k is present in the basket then there are initially |k| copies of it. The child pulls out the integers from the basket at random and arranges them in a sequence ak (a1 is the first integer the child pulls out, a2 is the second, and so on). He/she does not return the integers to the basket. Prove that the sequence bn = 1/an converges. What about a sequence cn = a(n+1) - an.
Homework Equations
The Attempt at a Solution
This is what my group and I came up with for our homework assignment - let us know what you think - thanks!
Given ε>0,choose a natural number M so that 1/M < ε. The set S = {k: |ak|<M} is finite, so max(S) exists. Let n = max(S) + 1. Then for all n > M, an does not belong to S, and hence |an| > M, and |bn|=|1/an| < 1/M < ε. Hence, bn converges to 0.
The sequence cn does not necessarily converge. For example, the basket could have exactly |k| copies of each integer k, and the sequence an could be the sequence of square an = n2. Then cn = 2n + 1, so the sequence diverges.