Since $f$ is continuous on $(a,b)$, bounded and increasing, there's a unique continous extension of $f$ to $[a,b]$. This works because both limits $f(b) := \lim_{x\to b-}$ and $f(a) = \lim_{x\to a+}$ and are guaranteed to exist since every bounded and increasing (respectively bounded a decreasing) sequence converges. To prove this, simply observe that for a increasing and bounded sequence, all $x_m$ with $m > n$ have to lie within $[x_n,M]$ where $M=\sup_n x_n$ is the upper bound. Add to that the fact that by the very definition of $\sup$, there are $x_n$ arbitrarily close to $M$.
You can then use the fact that continuity on a compact set implies uniform continuity, and you're done. This theorem, btw, isn't hard to prove either (and the proof shows how powerful the compactness property can be). The proof goes like this:
First, recall the if $f$ is continuous then the preimage of an open set, and in particular of an open interval, is open. Thus, for $x \in [a,b]$ all the sets $$
C_x := f^{-1}\left(\left(f(x)-\frac{\epsilon}{2},f(x)+\frac{\epsilon}{2}\right)\right) $$
are open. The crucial property of these $C_x$ is that for all $y \in C_x$ you have $|f(y)-f(x)| < \frac{\epsilon}{2}$ and thus $$
|f(u) - f(v)| = |(f(u) - f(x)) - (f(v)-f(x))|
\leq \underbrace{|f(u)-f(x)|}_{<\frac{\epsilon}{2}}
+ \underbrace{|f(v)-f(x)|}_{<\frac{\epsilon}{2}} < \epsilon
\text{ for all } u,v \in C_x
$$
Now recall that an open set contains an open interval around each of its points. Each $B_x$ thus contains an open interval around $x$, and you may wlog assume that its symmetric around $x$ (just make it smaller if it isn't). Thus, there are $$
\delta_x > 0 \textrm{ such that }
B_x := (x-\frac{\delta_x}{2},x+\frac{\delta_x}{2})
\subset (x-\delta_x,x+\delta_x)
\subset C_x
$$
Note how we made $B_x$ artifically smaller than seems necessary, that will simplify the last stage of the proof. Since $B_x$ contains $x$, the $B_x$ form an open cover of $[a,b]$, i.e. $$
\bigcup_{x\in[a,b]} B_x \supset [a,b] \text{.}
$$
Now we invoke compactness. Behold! Since $[a,b]$ is compact, every covering with open sets contains a finite covering. We can thus pick finitely many $x_i \in [a,b]$ such that we still have $$
\bigcup_{1\leq i \leq n} B_{x_i} \supset [a,b] \text{.}
$$
We're nearly there, all that remains are a few applications of the triangle inequality. Since we're only dealing with finitly many $x_i$ now, we can find the minimum of all their $\delta_{x_i}$. Like in the definition of the $B_x$, we leave ourselves a bit of space to maneuver later, and actually set $$
\delta := \min_{1\leq i \leq n} \frac{\delta_{x_i}}{2} \text{.}
$$
Now pick arbitrary $u,v \in [a,b]$ with $|u-v| < \delta$.
Since our $B_{x_1},\ldots,B_{x_n}$ form a cover of $[a,b]$, there's an $i \in {1,\ldots,n}$ with $u \in B_{x_i}$, and thus $|u-x_i| < \frac{\delta_{x_i}}{2}$. Having been conservative in the definition of $B_x$ and $\delta$ pays off, because we get $$
|v-x_i| = |v-((x_i-u)+u)| = |(v-u)-(x_i-u)|
< \underbrace{|u-v|}_{<\delta\leq\frac{\delta_{x_i}}{2}}
+ \underbrace{|x_i-u|}_{<\frac{\delta_{x_i}}{2}}
< \delta_{x_i} \text{.}
$$
This doesn't imply $y \in B_{x_i}$ (the distance would have to be $\frac{\delta_{x_i}}{2}$ for that), but it does imply $y \in C_{x_i}$!. We thus have $x \in B_{x_i} \subset C_{x_i}$ and $y \in C_{x_i}$, and by definition of $C_x$ (see the remark about the crucial property of $C_x$ above) thus $$
|f(x)-f(y)| < \epsilon \text{.}
$$
Some remarks concerning your proof:
You write:
Since $x \in X$ is an adherent point, then there exists a sequence $(a_n)$ such that $a_n \in X$ and converges to $x$.
Note that, by definition, we have to consider any two sequences $(a_n)_n$, $(b_n)_n$ in $X$ which are equivalent. This means in particular that $(a_n)_n$ need not be convergent and it does not suffices to prove the statement for some sequence $(a_n)_n$ (convergent to $x$), but we have to prove it for all sequences $(a_n)_n$ (equiavalent to some sequence $(b_n)_n$.)
You write:
Choose $\epsilon = \delta$.
In my oppinion, that's not a good formulation - both $\delta$ and $\epsilon$ are fixed numbers and we cannot simply set $\epsilon = \delta$. In this situation it is better two introduce a new variable in the following way:
Since $(a_n)_n$ and $(b_n)_n$ are equivalent, $$\forall \varrho>0 \, \exists N \, \forall n \geq N: |a_n-b_n| \leq \varrho.$$ Choose $\varrho = \delta$.
(a) $\Rightarrow$ (b) (Corrected version):
Let $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \mathbb{N}} \subseteq X$ be two equivalent sequences and $\epsilon>0$. Since $f$ is uniformly continuous, we can choose $\delta>0$ such that $$|x-y| < \delta \Rightarrow |f(x)-f(y)|< \epsilon. \tag{1} $$ It follows from the very definition of equivalent sequences that $$\forall \varrho>0 \, \exists N \in \mathbb{N} \, \forall n \geq N: |a_n-b_n| < \varrho.$$ Choose $\varrho = \delta$. Then $|a_n-b_n| < \delta$ implies $$|f(a_n)-f(b_n)| < \epsilon$$ by $(1)$ for all $n \geq N$. Since $\epsilon>0$ is arbitrary, this shows that $(f(a_n))_{n \in \mathbb{N}}$ and $(f(b_n))_{n \in \mathbb{N}}$ are equivalent.
(b) $\Rightarrow$ (a):
Suppose that $f$ is not uniformly continuous. Then there exists $\epsilon>0$ such that for any $\delta>0$, we can find $x,y \in X$ such that $$|x-y|< \delta \qquad \text{and} \qquad |f(x)-f(y)| \geq \epsilon.$$ Choosing $\delta = \frac{1}{n}$ for $n \in \mathbb{N}$, we find sequences $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ such that $$|x_n-y_n| \leq \frac{1}{n} \qquad \text{and} \qquad |f(x_n)-f(y_n)|> \epsilon.$$ Show that $(x_n)_{n \in \mathbb{N}}$ and $(y_n)_{n \in \mathbb{N}}$ are equivalent and that $(f(x_n))_{n \in \mathbb{N}}$, $(f(y_n))_{n \in \mathbb{N}}$ are not equivalent.
Best Answer
Your argument doesn't work. Consider the function $\frac{\sin\left(\frac1x\right)}x$. It is unbounded near $0$. However, if $a_n=\frac1{n\pi}$, then $\lim_{n\to\infty}f(a_n)=0$.
Here's a proof. There is a $\delta>0$ such that $\lvert x-y\rvert<\delta\implies\bigl\lvert f(x)-f(y)\bigr\vert<1$. Now, consider points $a_1,a_2,\ldots,a_N\in(a,b)$ such that $a<a_1<a_2<\cdots<a_N<b$ and that $a_1-a,a_2-a_1,\ldots,b-a_M<\delta$. Then, if $x,y\in(a,b)$ and $x<y$, there are numbers $k<l\leqslant N$ such that $a_{k-1}<x\leqslant a_k$ and $a_{l-1}<y\leqslant a_l$. But then\begin{align}\lvert f(x)-f(y)\bigr\vert&\leqslant\overbrace{\bigl\lvert f(x)-f(a_k)\bigr\rvert}^{<1}+\overbrace{\bigl\lvert f(a_k)-f(a_{k+1})\bigr\rvert}^{<1}+\cdots+\overbrace{\bigl\lvert f(a_l)-f(y)\bigr\rvert}^{<1}\\&\leqslant l-k+1.\end{align}