A compact locally connected metric space is "uniformly locally connected" A compact locally connected metric space is "uniformly locally connected"\
That is, for any $\epsilon > 0$, there is some $\delta > 0$ such that whenever $\rho(x, y) < \delta$, then $x$ and $y$ both lie in some connected subset of $X$ of diameter $<\epsilon$.
proof:-
Since $X$ is locally connected metric space then each $x\in X$ has a nhood base of open connected sets\
Given $\epsilon > 0$, let $x \in X $ and $U_x=\rho(x, \epsilon) $ be a nhood of $x$\
There exist an open connected basic nhood $V_x$ with diameter $<\epsilon$,
Now $$X=\bigcup_{x\in X }{V_x}$$, hence cover $X$ by open connected nhoods of diameter $<\epsilon$.\
Since $X$ is compact, reduce this to a finite subcover $\{V_{x1},. . . , V_{xn}\}$ and let $\delta$ be a Lebesgue number (22.5) for
this cover.\
Then if $\rho(x, y) < \delta$, both $x$ and $у$ belong to some $V_{xi}$.
\
{Theorem 22.5} (Lebesgue covering lemma). If $\{U_1..., U_n\}$ is a finite open
cover of a compact metric space X, there is some $\delta > 0$ such that if A is any
subset of $X$ of diameter $< \delta$, then $A \subset U_i$ for some i.
I try to write the proof better than this.

I would like to confirm this proof
If acceptable, I would like to clarify and improve it (Language and Mathematical)as much as possible

 A: The proof is correct. 
As for writing it up. That depends on your audience. For a general audience, the first rule is - avoid abbreviations ("neighborhood", not "nhood"). And I assume that the "/" scattered through the text was some sort of misguided attempt to show where you put new lines? Just let Latex have it's way here. If a new line is needed, let it form a paragraph break.
Also "and $U_x=\rho(x,\epsilon)$ be a nhood of $x$" doesn't make sense. $\rho$ is a function that takes points in your space in both arguments, and produces real numbers. You have it with a real number as the second argument, and producing a set. Now it is evident that you mean $U_x$ to be a ball of radius $\epsilon$ about $x$. But if someone has told you this is a good way to denote the ball, do NOT trust them about notations in the future! Commonly used ways of denoting balls include $B(x, \epsilon)$ and $B_x(\epsilon)$. If the metric to be used is not obvious, then usually $B_\rho(x, \epsilon)$ is preferred. (Of course, it you had intended $B(x, \epsilon)$ and only accidently used $\rho$, just be careful in your final write-up.)
But you don't do anything with "$U_x$", which makes introducing a notation for it pointless. You don't actually need to name the balls, since you don't plan on using that name.
There are also some phrasing differences I would suggest. Your phrasing is clear enough, but sounds awkward sometimes to a native English speaker (at least - to this native English speaker). There is nothing particularly bad, though.
I would write it up like this:

Since $X$ is locally connected, for $\epsilon > 0$, each point $x \in X$ has a open connected neighborhood $V_x$ contained within the ball of radius $\epsilon$ about $x$. The collection $\{V_x \mid x \in X\}$ forms an open cover of $X$ by sets of diameter $< \epsilon$. Since $X$ is compact, it has a finite subcover $\{V_1, V_2, ..., V_m\}$. By the Lesbegue covering lemma (Theorem 22.5), there is a $\delta > 0$ such that if $\rho(x, y) < \delta$, then there is some $V_i$ with $\{x, y\} \subset V_i$, which completes the proof.

