I've personally found things are a lot easier to prove using the sequences rather than the $\epsilon-\delta$ method. I do understand the $\epsilon-\delta$ is more intuitive, but it's quite difficult to prove a function is continuous on $x_0 \in D$ since it implies finding a function that takes $\epsilon$ and $x_0$ as arguments and outputs some $\delta>0$, or at least proving such a function exists. This issue is even more prevalent when proving a function is uniformly continuous. On all of my homework problems involving uniform continuity, I always tried to use the $\epsilon-\delta$ definition, yet I always failed. When searching for the solution, the proofs are quite confusing and complex. I would then switch to the sequence definition and would succeed in solving the problem quickly. My professor says that the mathematics community nonetheless refers the $\epsilon-\delta$ definition, so I am curious why when it's a more difficult path.
Here are the definitions.
For a function $f:D \rightarrow \mathbb{R}$,
Continuity:$f$ is continuous at $x_0 \in D$ if for any sequence $(x_n)$ in $D$ where $\lim_{n \rightarrow \infty} x_n=x_0$, it follows $\lim_{n \rightarrow \infty} f(x_n)=f(x_0)$.
Uniform continuity: $f$ is uniformly continuous if for any sequences $(u_n),(v_n)$ in $D$ where $\lim_{n \rightarrow \infty} (u_n-v_n)=0$, it follows $\lim_{n \rightarrow \infty}(f(u_n)-f(v_n))=0$ .
An instance of this is proving the function $f:[0,1) \rightarrow \infty$ where $f(x)=\frac{1}{1-x}$ is not uniformly continuous. I found it very difficult to prove this using the $\epsilon-\delta$ definition, and the proofs I found online were quite confusing and there's definitely no way I would've come across them by myself. The sequential definition however was very simple. I simply used the sequences $u_n=1-\frac{1}{n^2}$ and $v_n=1-\frac{1}{n}$. The differences of the sequences converge to 0, and the differences of the image of the sequences diverge to $\infty$. $0 \not = \infty$ and therefore $f$ is not uniformly continuous.
Another example is showing that any continuous function whose domain is a closed interval is uniformly continuous. There's no way I could come up with a way to prove this using the $\epsilon-\delta$ definition, but I definitely could using the sequence definition.
I may be wrong, but I believe I read once that for some metric spaces, continuity and sequential continuity are not equivalent. I don't know much about that matter though, but would love to learn about it.