Proof of an Infimum of a Subset

Question

Let $B$ be a subset of $\mathbb{R}$ and $B\neq\emptyset$ is bounded from below. For $y\in\mathbb{R}$, $y-\frac{1}{n}$ is a lower bound of $B$, but $y+\frac{1}{n}$ is not a lower bound of $B$, for any $n\in\mathbb{N}$. Show that $\inf B=y$.

Answer 1

I have attempted to answer this, but I got stuck in the middle. Since $B$ is bounded from below and, if $y\in\mathbb{R}$, then $y-\frac{1}{n}$ is a lower bound of $B$ for any $n\in\mathbb{N}$, then $\forall_{b\in B},b\geq y-\frac{1}{n}$. However, $y+\frac{1}{n}$ is not a lower bound of $B$ so it is safe to say $\forall_{b\in B},b<y+\frac{1}{n}$. Thus, we have discovered that $\forall_{b\in B},y-\frac{1}{n}\leq b<y+\frac{1}{n}$. How should I continue this?

Answer 2

$B$ is bounded from below, so it has a greatest lower bound. Is it possible for the GLB to be less than $y$? Can it be greater than $y$?