A typical basic open nbhd of $h(t)$ in the uniform topology is $B(h(t),r) = \prod_{k\in\mathbb{Z}^+}\left(\frac{t}{k}-r,\frac{t}{k}+r\right),$ where $r$ is any positive real number.
Correction: This isn’t quite correct: the sequence $\left(\frac{t}{k}+\frac{k-1}{k}r\right)_{k\in\mathbb{Z}^+}$ is in the product of open intervals but not in the open ball of radius $r$ about $h(t)$. The correct definition: $B(h(t),r) = \bigcup_{0<\epsilon
For $s\in\mathbb{R}$, $h(s) \in B(h(t),r)$ if and only if there is a positive $\epsilon such that $\frac{s}{k} \in \left(\frac{t}{k}-\epsilon,\frac{t}{k}+\epsilon\right)$ for each $k \in \mathbb{Z}^+$. What happens if $|s-t|? (I’ll expand this to a complete solution if you get completely stuck, but you should first try to finish it from here.)
Added: Let’s look at a specific example: the nbhd $B\left(h\left(\frac34\right),\frac14\right)$ of $h\left(\frac34\right)=\left(\frac34,\frac38,\frac3{12}\dots\right)$, the sequence whose $k$-th term is $\frac3{4k}$.
Is $\frac12 \in h^{-1}\left[B\left(h\left(\frac34\right),\frac14\right)\right]$? No: $h\left(\frac12\right)= \left(\frac12,\frac14,\frac16,\dots\right)$, and $\frac12 \notin \left(\frac34-\epsilon,\frac34+\epsilon\right)$ for any $\epsilon<\frac14$. What about $\frac58$? $h\left(\frac58\right) = \left(\frac58,\frac5{16},\frac5{24},\dots\right),$ the sequence whose $k$-th term is $\frac5{8k}$. Is it true that there is a positive $\epsilon<\frac14$ such that $\frac5{8k} \in \left(\frac3{4k}-\epsilon,\frac3{4k}+\epsilon\right)$ for every positive integer $k$? This is the same as asking whether $\frac3{4k}-\epsilon<\frac5{8k}<\frac3{4k}+\epsilon$ for every $k\in\mathbb{Z}^+$. Multiply through by $8k$ and subtract $5$to get $1-8k\epsilon<0<1+8k\epsilon$; as long as $\epsilon>\frac18$, this is true for every positive integer $k$, so $\frac58$ is in $h^{-1}\left[B\left(h\left(\frac34\right),\frac14\right)\right]$.
Now try it in general. Suppose that $|s-t|; is $h(s)\in B(h(t),r)$?
Since $h(s)=(s,s/2,s/3,\dots)$, this is simply asking whether it’s true that there’s a positive $\epsilon such that $\frac{s}{k} \in \left(\frac{t}{k}-\epsilon,\frac{t}{k}+\epsilon\right)$ for all $k\in\mathbb{Z}^+$, i.e., whether $\frac{t}{k}-\epsilon<\frac{s}{k}<\frac{t}{k}+\epsilon\;.$ What happens if you pick $\epsilon\in(r-|s-t|,r)$?