Problem in understanding big-O notation and similiar
$O(n)$ is the set of all functions $f(n)$ such that, for sufficiently large $n$ and a positive constant $c$, $f(n) \le cn$.
$1+\Theta(\frac{1}{n})$ is an abuse of notation that denotes the set of all functions $h(n) = 1+g(n)$ such that, for sufficiently large $n$ and a two positive constants $c_1$, $c_2$, $\frac{c_1}{n} \le g(n) \le \frac{c_2}{n}$.
Then the claim is clearly false since, e.g., the function $h(n)=n$ belongs to $O(n)$ but it does not belong to $1+\Theta\left(\frac{1}{n}\right)$.