Relationshop between Theta and little-o (Big-O notation)
To expand slightly on Karolis's comment, you can get a counterexample by taking for any strictly positive function $f$ and, for any constant $\delta&gt;0$, the function $(1+\delta)f(x)$.  $(1+\delta)f(x)\in \Theta(f(x))$ but $(1+\delta)f(x)\notin (1+o(1))f(x)$ since, for $\epsilon&lt;\delta$, it is not true that $(1+\delta)f(x)\leq (1+\epsilon)f(x)$.