I believe I have the right answers for these two questions, but any corrections/clarifications would be very helpful. The two questions, I'm looking at are:

1) Prove that the distance function $\displaystyle d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ defined as $\displaystyle d(\mathbf{x},\mathbf{y})= |\mathbf{x}-\mathbf{y}|$ is continuous. Use the two versions of the triangle inequality.

For this one I proved it using Epsilon=Delta. Is this correct?

2) Prove that the dot product in $\displaystyle \mathbb{R}^n$ is continuous. Use Cauchy-Schwartz inequality.

Again, I believe I proved this using Epsilon=Delta. Is this correct as well?

Thanks a lot