Part 1) Let $g_n:[0,1]\to R $ be a sequence of uniformly continuous functions which converges uniformly to a function $g:[0,1]\to R$. Prove that $g$ is uniformly continuous.
Part 2) Let $g_n:(0,1)\to R $ be a sequence of uniformly continuous functions which converges uniformly to a function $g:(0,1)\to R$. Prove that $g$ is uniformly continuous.
The second part is just a variant of the first, except with an open interval instead of a closed one. Does anyone know of a way prove both of these rigorously?
Hint: Apply $\epsilon/3$-argument to the inequality $|g(x)-g(y)|\leq|g(x)-g_{n}(x)|+|g_{n}(x)-g_{n}(y)|+|g_{n}(y)-g(y)|$.