Proving Riemann Integrability of continuous function over compact interval

$$\textbf{The Problem:}$$ Let $$f:[a,b]\to\mathbb R$$ be a continuous function. Prove that $$f$$ is Riemann integrable on $$[a,b]$$.

$$\textbf{My Thoughts and Attempt:}$$ Since $$[a,b]$$ is compact, $$f$$ is also uniformly continuous on it, so there is a $$\delta(\varepsilon)>0$$ such that for all $$\varepsilon>0$$ and all $$x,y\in[a,b]$$ with $$\vert x-y\vert\leqslant\delta(\varepsilon)$$ it follows that $$\vert f(x)-f(y)\vert<\varepsilon.$$ So let $$\varepsilon>0$$ be given and consider the partition $$\mathcal P=\{x_0=a,x_1=a+\delta(\varepsilon), x_2=a+2\delta(\varepsilon),\dots,a+(n-1)\delta(\varepsilon),x_n=b\},$$ where $$n\in\mathbb N$$ is such that $$\vert x_j-x_{j-1}\vert\leqslant\delta(\varepsilon)$$ implies $$\color{red}{\vert f(x)-f(y)\vert<\displaystyle\frac{\varepsilon}{n}}$$. Then the uniform continuity of $$f$$ guarantees that the values $$M_j=\sup\limits_{x\in[x_{j-1},x_j]}f(x)\quad\text{and}\quad m_j=\inf\limits_{x\in[x_{j-1},x_j]}f(x)$$ are attained and that $$\color{red}{M_j-m_j<\displaystyle\frac{\varepsilon}{n}}$$. So we see that \begin{align*} \mathcal U(f,\mathcal P)-\mathcal L(f,\mathcal P)&=\sum^{n}_{j=1}(x_{j}-x_{j-1})\cdot(M_j-m_j)\\ &\leqslant\delta(\varepsilon)\cdot\sum^{n}_{j=1}\frac{\varepsilon}{n}\\ &=\delta(\varepsilon)\cdot\varepsilon. \end{align*} And it follows that $$f$$ is Riemann integrable on $$[a,b]$$.

$$\textbf{Note:}$$ The notation $$\delta(\varepsilon)$$ means that $$\delta$$ can only depend on $$\varepsilon.$$

$$\textbf{My Concern:}$$ My concern is that I am not sure if the statements in $$\color{red}{\text{red}}$$ are correct. Also, if there are any other mistakes, please feel free to point them out.

Thank you for your time.

No it should be $$M_{j} - m_j \leq \varepsilon.$$ You need just proceed like following \begin{align*} \mathcal U(f,\mathcal P)-\mathcal L(f,\mathcal P)&=\sum^{n}_{j=1}(x_{j}-x_{j-1})\cdot(M_j-m_j)\\ &\leqslant\varepsilon\sum^{n}_{j=1} (x_{j}-x_{j-1})\\ &= (b-a )\varepsilon. \end{align*}