Proposition: Let $X$ be a subset of $R$, let $f:X\to R$ be a uniformly continuous function, and let $x_0$ be an adherent point of $X$. Then $\lim_{x\to x_0 ;x\in X} f ( x)$ exists.
Proof Take any sequence $(a_n)_{n=1}^\infty$ where $a_n\in X$ and it converges to $x_0$. We know such a sequence exists because $x_0$ is an adherent point of $X$.
Since $(a_n)$ is a convergent sequence, then its cauchy. $f$ is a uniform continuous function,hence, if $(a_n)$ is cauchy, $(f(a_n))$ is also cauchy. Again, because $(f(a_n))$, then its convergent and hence the limit exists.
Is my proof correct?
You have just proved limit exists for a specific convergent sequence $x_n\to x_0$, you have to show then the limit is independent of the convergent sequence. That is if $x_n\to x_0,y_n\to x_0$, then $\lim f(x_n)=\lim f(y_n)$, which should be easy if you make use of the uniform continuity of $f$ on $X$ and pass to limit.
Respond to J.G's comment:
Given $x_n\to x_0,y_n\to x_0$, we know $\lim f(x_n),\lim f(y_n)$ exists and aim to show $\lim f(x_n)=\lim f(y_n)$. Consider sequence $z_n$ defined as $(x_1,y_1,x_2,y_2,\cdots)$ which is a combination of $(x_n),(y_n)$. Then we know $z_n\to x_0$ and by similar argument, we know $\lim f(z_n)=l\in \mathbb{R}$. Since $f(x_n),f(y_n)$ are two subsequence of a convergence sequence $\lim f(z_n)$, we know they must convergent to the same limit, hence $\lim f(x_n)=\lim f(y_n)$.