Suppose that $X, Y$ are random variables. My aim is to show that $X\overset{d}{\to} N(0,\sigma^2)$. If I assume that $X-Y=o_p(1)$ and $Y\overset{d}{\to} N(0,\sigma^2)$, is it right to conclude that $$X=(X-Y)+Y=o_p(1)+Y\overset{d}{\to} N(0,\sigma^2),$$ using Slutsky's Theorem?

*$o_p(1):=$ convergence in probability to zero.


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