Question

Prove that a non-decreasing (resp. non-increasing) sequence which is not bounded above (resp. bounded below) diverges to +∞ (resp. to −∞)

Answer 1

If a non-decreasing sequence which is not bounded above, then for each value $C>0$ there exists $a_n$ such that $a_n>C$ . So, for $C\to+\infin$ there exists $a_n>C$ . That is, $(a_n)\to+\infin$