Homogeneous generators of a homogeneous ideal

Let $k$ be an algebraically closed field, and $I\subset k[x_0,\cdots,x_n]$ a homogeneous ideal of height $r$. If $I$ can be generated by $r$ elements, can we pick $r$ homogeneous elements in $I$ that generate $I$?

Background: Let $Y=V_+(I)$ be the corresponding closed subscheme. $Y$ is called a complete intersection if $I$ can be generated by $r$ elements. But it sounds weird to me, since it doesn't ask the generators to be homogeneous, which is not convenient to consider their corresponding hypersurfaces. So I guess we may choose $r$ homogeneous generators for $I$.

Could you provide some help?(prove it or give an counterexample) Thanks!

Suppose $I$ is generated by $r$ elements. Let $\{f_s\}_{s\in S}$ be a minimal system of homogeneous generators for $I$. Any relation $\sum g_sf_s=0$ among the $f_s$ must have $g_s\in(x_0,\cdots,x_n)$: if some $g_{s_0}$ has a nonzero constant term, then looking at the $\deg f_{s_0}$ component of the equation, we see that $f_{s_0}$ is in the ideal generated by the other $f_s$ involved. Now define $I\to k^{\oplus S}$ by $\sum g_sf_s=(g_s(0))_{s\in S}$: this is surjective and does not depend on the representation of an element as $\sum g_sf_s$ by the previous sentence. This implies that $k^{\oplus S}$ can be generated by $r$ elements, or $r\geq |S|$. Since every generating set of $I$ must have at least $\operatorname{ht} Y$ elements, we see that $I$ can indeed be generated by $r$ homogeneous elements.