hint, for first question, please prove it is wrong, do not pick up a particular example to prove it is wrong second questyion, hint, u can approach question by using subsequential limit stuffs

1. We have xn = (−1)n + 1n , suppose a 6= ±1, we prove that there does not
exists a subsequence xnk of xn such that xnk → a.
Suppose not, that is suppose there is subsequence xnk such that xnk → a.
Hence for given � > 0 there exists N such that
(−1)nk + 1
nk
∈ (a− �, a+ �); for all nk ≥ N
That is we...