Hey guys,

Can you check whether my proof of the problem?

Show that a monotone sequence has at most one cluster point.

pf)

i) {a_n} is monotone increasing and bounded above.

By monotone convergence theorem, {a_n} -> x

If {a_n} is monotone decreasing and bounded below

b_n = -a_n and apply the same as above.

a_n -> x

Since a_n converges to x, a_n only has x as its cluster point.

ii) {a_n} is monotone increasing and not bounded.

There is N in N such that |a_N| $\displaystyle \geq M, \forall M>0 $

Assume a_n -> x.

For any epsilon > 0 There is a_N such that |a_N - x | < 1, when n>N.

|a_N| - |x| $\displaystyle \leq |a_N - x| < 1 $

M = Max { |a_1|, |a_2|, ... |a_N|, |a_n| +1}

|a_n| $\displaystyle \leq M $

Since a_n is unbounded, contradiction.

Since a_n is unbounded and diverges, a_n has no cluster point.

If a_n is monotone decreasing and unbounded, let b_n = - a_n and apply the same procedure.