Consider $[n]=\{1, 2, \cdots, n\}$ with the discrete topology and let $X=\prod\_{n\geq 1}[n]$ be the product space with the product topology. For $x=(a\_1, a\_2, \cdots)$, define $T(x)=(1, a\_1, a\_2, \cdots)$. Then how to show the following?

1. The map $T:X\to X$ has a unique fixed point
2. The map $T:X\to X$ is continuous

For part 1, let $T(x)=x$, then $(a\_1, a\_2, \cdots)=T(x)=(1, a\_1, a\_2, \cdots)$ which means $a\_1=1$, $a\_2=a\_1=1$, $a\_3=a\_2=1, \cdots$. Therefore, $x=(1, 1, 1, \cdots)$ is the unique fixed point.

But I don't know how to prove the second part. Please help how to initiate? Thanks in advance.