If \(s_1\), \(s_2\), \(s_3\), \(\ldots\) and \(t_1\), \(t_2\), \(t_3\), \(\ldots\) are sequences of positive numbers, we write
\[
(s_n)\le (t_n)
\]
to mean
"there exists a positive integer \(m\) such that \(s_n \le t_n\) whenever \(n\ge m\)".
Determine whether each of the following statements is true or false. In the case of a true statement, you should give a proof which includes an explicit determination of an appropriate \(m\); in the case of a false statement, you should give a counterexample.
\((1000n) \le (n^2)\,\).
If it is not the case that \((s_n)\le (t_n)\), then it is the case that \((t_n)\le (s_n)\,\).
If \((s_n)\le (t_n)\) and \((t_n) \le (u_n)\), then \((s_n)\le (u_n)\,\).
\((n^2)\le (2^n)\,\).
Solution:
If \(m = 1000\), then \(n \geq m \Rightarrow n^2 \geq 1000n \Rightarrow (1000n) \leq (n^2)\)
This is false. Let \(s_i = 1,2,1,2,\cdots\) and \(t_i = 2,1,2,1,\cdots\).
Suppose that for \(n \geq m_1, s_n \le t_n\) and for \(n \geq m_2, s_t \le u_n\), then for \(n \geq m = \max(m_1, m_2), s_n \leq t_n \leq u_n \Rightarrow s_n \leq u_n \Rightarrow (s_n) \leq (u_n)\)
Let \(m = 6\), then if \(n \geq m, 2^n \geq 1 + n + \frac{n(n-1)}{2} + \frac{n(n-1)}{2} + n + 1 = n^2 + n + 2 \geq n^2\), so \((2^n) \geq (n^2)\)