Is the sequence with limit necessarily convergent? Is it not necessarily bounded

Is the sequence with limit necessarily convergent? Is it not necessarily bounded

Is the sequence with limit necessarily convergent: Yes
Isn't there a limit to being bounded: Yes
e.g
|Sin (1 / x) | 0) sin (1 / x) does not exist

Proving the convergence of sequence and finding the limit in Higher Mathematics
Let A1 = 1, when n > = 1, a (n + 1) = (an / 1 + an) ^ 1 / 2, prove the convergence of sequence and find its limit

A (n + 1) = [an / (1 + an)] ^ (1 / 2) | an | > 0 {an} decreasing = > LIM (n - > ∞) an existslim (n - > ∞) a (n + 1) = LIM (n - > ∞) [an / (1 + an)] ^ (1 / 2) l = (L / (1 + L)) ^ (1 / 2) L ^ 2 (1 + L) = LL (L ^ 2 + L - 1) = 0l = (- 1 + √ 5) / 2lim (n - > ∞) an = L = (- 1 + √ 5) / 2

The problem that convergence sequence must be bounded
Isn't it necessary for a bounded sequence to have upper and lower bounds? Isn't it necessary for a convergent sequence to have both upper and lower bounds

Yes, the convergent sequence must be bounded, but not both upper and lower bounds. Being bounded is a necessary condition for the existence of a limit, but being bounded does not necessarily mean that there is a limit