# Convergence in distribution: proof strategy verification (asymptotic normality)

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.