$\mathbb{R^\omega}$ is normal in product topology and uniform topology. To prove: $\mathbb{R^\omega}$ is normal in product topology and uniform topology.
One theorem says that Every Metrizable Space is Normal. So if we can show that $\mathbb{R^\omega}$ is metrizable with product and uniform topology then we are done.
But I am facing problem in showing it....Help Needed!

Let define $|\cdot |_b :\Bbb{R}\to \Bbb{R}$ as
$$|x|_b=\begin{cases} |x| & \text{if }|x|\le 1 \\ 1 & \text{otherwise}\end{cases}$$
and consider the metric
$$d(x,y) = \sup_{n\in \Bbb{N}} \frac{|x_n-y_n|_b}{n}$$
and
$$\rho(x,y) = \sup_{n\in \Bbb{N}} |x_n-y_n|_b$$
for $x=(x_n)$ and $y=(y_n)$. You can check that $d$ induces product topology and $\rho$ induces uniform topology.
