Let (an)n ≥ 1 be a sequence of positive real numbers. Let \(\rm b_n=\frac{a_n}{max\{a_1..., a_n\}}n \ge 1 \)
Which of the following statements are necessarily true?
1
If \(\rm \lim_{n \rightarrow \infty}b_n\) exists in ℝ, then {an : n ≥ 1} is bounded
2
If \(\rm \lim_{n \rightarrow \infty}b_n=1\) then \(\rm \lim_{n \rightarrow \infty}a_n\) exists in ℝ
3
If \(\rm \lim_{n \rightarrow \infty}b_n=\frac{1}{2}\), then \(\rm \lim_{n \rightarrow \infty}a_n\) exists in ℝ
4
If \(\rm \lim_{n \rightarrow \infty}b_n=0\), then \(\rm \lim_{n \rightarrow \infty}a_n=0\)