A course in real analysis by Hugo D. Junghenn

By Hugo D. Junghenn

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Notation. It is occasionally convenient to use the following alternate method to describe a subsequence: If we set bk = ank and then change the index in {bk }∞ k=1 to n, then {bn } may be used to denote the subsequence {ank }. This provides a convenient way to denote a subsequence of a subsequence. In this regard, note that if {cn } is a subsequence of {bn } and {bn } is a subsequence of {an }, then {cn } is a subsequence of {an }. The following proposition shows that a convergent sequence has a single cluster point.

The sets (a, +∞), and (a, +∞) ∩ Q, a < 1, are clearly inductive. More importantly, N itself is inductive. Indeed, since 1 is common to all inductive sets, 1 ∈ N, and if n is common to all inductive sets, then so is n + 1. We may therefore characterize N as the smallest inductive set (in the sense of set inclusion). 2 Principle of Mathematical Induction. For each n ∈ N, let P (n) be a statement depending on n. Suppose that (a) P (1) is true, (b) P (n + 1) is true whenever P (n) is true. Then P (n) is true for all n.

Bs The following proposition gives a simple way to generate irrational numbers. 11 Proposition. If n is positive integer that is not a perfect square, then √ n is irrational. √ √ √ Proof. √By definition of the greatest integer function, n − 1 < n ≤ n. Since n is is strict, √ √ assumed √ not to be an integer, the second inequality hence 0 < n − n < 1. Suppose, for a contradiction, that n is rational. √By the well-ordering principle, A has a least member m0 . In particular, m0 n ∈ N, hence both of the quantities √ √ √ √ √ m := m0 n − n and m n = m0 n − n n are positive√integers.

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