For example I know that $A=\{5-\frac{1}{n}\}_{n \in \mathbb{N}}\cup\{6,7,8,...,k+6\}$ is isomorphic to $\omega + k$ for $k<\omega\;$ because I know I can look at $\omega + k\;$ as $\; B=(n\times\{0\})_{n\in\mathbb{N}}\cup(j\times\{1\})_{6\leq j\leq k+6} $ with the lexicograpic order.
However, If I wanted to find real numbers subset which is isomorphic to $\omega + \omega + k\;$ for $k<\omega\;$
How could I do that? how can I visualize this ordinal in order to give it a mataching real subset.
As well as for $\omega + \omega + \omega + ... + \omega$ (n times)
I would love to get some examples and intuition if possible! thanks.
If $\alpha$ is a countable ordinal, then there is an injective map $f:\alpha\to\omega$, and $$\xi\mapsto\sum_{\eta\lt\xi}2^{-f(\eta)}$$ is an order embedding of $\alpha$ into $\mathbb R$.
A similar argument shows that every countable linear order is embeddable in $\mathbb R$. A different argument is needed if you want to show that every countable linear order is embeddable in $\mathbb Q$, or that every countable ordinal is order-isomorphic to a closed subset of $\mathbb R$.