Let’s start by recalling an important theorem of real analysis:
THEOREM. A necessary and sufficient condition for the convergence of a real sequence is that it is bounded and has a unique limit point.
As a consequence of the theorem, a sequence having a unique limit point is divergent if it is unbounded. An example of such a sequence is the sequence un=n2(1+(−1)n), whose initial values are 0,1,0,2,0,3,0,4,0,5,6,… (un) is an unbounded sequence whose unique limit point is 0.
Let’s now look at sequences having more complicated limit points sets.
A sequence whose set of limit points is the set of natural numbers
Consider the sequence (vn) whose initial terms are 1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,… (vn) is defined as follows vn={1 for n=1n–k(k+1)2 for k(k+1)2<n≤(k+1)(k+2)2 (vn) is well defined as the sequence (k(k+1)2)k∈N is strictly increasing with first term equal to 1. (vn) is a sequence of natural numbers. As N is a set of isolated points of R, we have V⊆N, where V is the set of limit points of (vn). Conversely, let’s take m∈N. For k+1≥m, we have k(k+1)2+m≤(k+1)(k+2)2, hence uk(k+1)2+m=m which proves that m is a limit point of (vn). Finally the set of limit points of (vn) is the set of natural numbers.
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