Prove that the closed interval [0,1] is a closed set and that the open interval (0,1) is an open set..

Could I just say that for [0,1], every open ball B(0,r), r > 0 contains at least one point less than 0 and therefore not an element of [0,1] to prove it is closed

and for (0,1), its complement (-inf, 0] U [1, inf) must be closed by similar reasoning?

is there a way to incorporate the theorem:

a set S is closed iff for all sequences {x_{k}} such that x_{k}is an element of S for each k and x_{k}--> x, it is the case that x is an element of S

I feel like Im basically just stating the definition of closed/open sets up top, and I don't know if that's an acceptable way to do it

Thanks