Verification of sequence result

Question

Is it true that if a real sequence $\{x_n\}_1^\infty$ has an infimum but no convergent subsequences then the infimum must be the minimum as well?

Secondly, can it be proved that the sequence defined as $x_n = (-1)^{n} + \sqrt{n}$ is a sequence which has no convergent subsequence?

Thanks.

Answer 1

Hint: This is indeed the case. Try to prove it by assuming the sequence has a infimum that is not a minimum as well, and show that the sequence has a convergent subsequence.