In my article, I would like to write the following sentence:
There exisits a vertex in $H$, without loss of generality, let it be $v$, such that $H-v$ is connected.
I think the sentence above is incorrect as two sentences connected by a comma. So I changed it to：
There is a vertex in $H$; let it be $v$ for the sake of generality, such that $H-v$ is connected.
But I feel that the sentence is not fluent. How can I make my sentence more fluent? I try this:
There is a vertex in $H$, namely $v$, such that $H-v$ is connected.
But this does not seem to reflect the meaning of "without loss of generality".
Edits:
PrincessEev's advice is nice. But if I had named $V(H)$ in advance and then wanted to choose $w_1$ without loss of generality, it would have seemed abrupt for me to write that below.
Let $V(H)$ be $\{v_1,v_2,\cdots,v_n\}$. There exists $v_1∈V(H)$ such that...
Or:
"There is a vertex (call it $v_1$) in $H$ such that..."