I am totally comfortable and can rigorously prove the fact that for any pair of random sequences say $\{X_n, Y_n\}_{n\in\mathbb{N}}$ I have: $X_n o_p(Y_n) = o_p(X_n Y_n)$, and in particular I also have $X_n o_p(1) = o_p(X_n)$. However one of the "basic facts" is that $o_p(X_n) = X_n o_p(1)$, and I am totally stuck on trying to prove this to myself using the definition/first principles.
If anyone could give a rigorous proof of this it would be greatly appreciated.
I think the point at which I get really confused is what it even means for a random sequence sequence $\{Z_n\}_{n\in\mathbb{N}}$ to be in the set of random sequences $X_n o_p(1)$.
Also I was curious as to whether or not this "basic fact" could be extended to the statement $o_p(X_n Y_n) = Y_n o_p(X_n)$ for any two general random sequences.
We say that $Y_n = o_p(X_n)$ if and only if
$$ \frac{Y_n}{X_n} \overset{p}{\to} 0. $$
This is equivalent to saying that $Y_n = X_nZ_n$ for some $Z_n \overset{p}{\to} 0$. To see this, just define
$$ Z_n = \frac{Y_n}{X_n}.$$
In other words, $Y_n = X_n o_p(1)$. Thus, $o_p (X_n) = X_n o_p (1)$.