Im having quite a problem with solving these just cuz I'm not sure what to do when I have a random real number a interacting with the sequence instead of another one.
1.) For a convergent real sequence sn and a real number a, show that if sn >= a for all but finitely many values of n, then lim n—>∞ Sn>=a 2.)For each value of a € R, give an example of a convergent sequence sn with sn > a for all n, but where lim n—>∞ Sn=a
For second part take $s_n=a+\frac 1 n$.
For the first part you have to assume that the limit exists. Suppose $l=\lim s_n$. If possible, let $l<a$. Then $|s_n-l| <\frac {a-l} 2$ for all $n$ exceeding some $n_0$. I will leave to you to to verify that $s_n <a$ for all $n > n_0$ which contradicts the hypothesis.