real analysis - How to prove $B$ is the least possible upper bound?

I saw the following sequence-

$S_n=\frac12,\frac23,\frac34,\frac45,\frac56,\ldots,\frac{n-1}{n}$.

Now, I may say that-

$$(A)\leq S_n\leq(B),$$

where, $A=\frac12$ and $B=1$

Now, am I correct in saying that $B$ is the least possible upper bound?

If yes, how do I prove it?

Thanks for any help.