In class, we defined initial segment embedding in the following way:
Initial segment embedding is a function $f:A\to B$ $s.t.$ $x\leq y\Longleftrightarrow f(x)\leq f(y)$, and range($f$) is an initial segment of $B$, $i.e.$ let $X=$range($f$), then $\forall y\forall x(x\in X, y\in B,y<x\Longrightarrow y\in X)$.
Here is my question: Let $b_0$ be the smallest element in $B$. Since we know that $\leq$ is reflexive, we have $b_0\leq b_0$. Then is the function $f:A\to B$, $f(x)=b_0$ an initial segment embedding?
Thoughts: I think it is since $x\leq y\Longleftrightarrow f(x)=b_0\leq b_0=f(y)$ and $\{b_0\}$ is the entire range of $f$ and it is clearly an initial segment following the definition.
Thank you!