Suppose that {xn} is a sequences such that every subsequence {xni} has a subsequence {xnmi} that converges to x. Show that {xn} is bounded.
I tried to do a proof by contradiction but am not sure if the proof is sufficient.
I said: Suppose {Xn} is unbounded.
Then, ∀ M∈R s.t. ∀ n∈N ∃ |Xn| > M.
Thus, ∀ subsequences that contain the set of n's we have that ∀ M∈R s.t. ∀ i∈N ∃ |Xni| > M.
Therefore, ∀ subsequences that contain the set of i's we have that ∀ M∈R s.t. ∀ m∈N ∃ |Xnmi| > M.
But, since we know {Xnmi} converges to x the subsequence cannot be unbounded.
Therefore, {Xn} must be bounded.
Is this proof sufficient. Any corrections or suggestions you have is appreciated.