Question: Let $R$ be a ring and let $J$ be a left ideal of $R$.
Assume that $J$ is nilpotent. Prove that $J$ is contained in a nilpotent 2-sided ideal of $R$.
Comments: I have found lots of solutions online which go through this proof but can't fully understand any of them. I want to try and use the fact that $J +JR ⊆ JR$.
One proof online stated the following: "It is easy to see that $J^n = 0$ implies $(J + JR)^n = 0$"
I can't seem to figure out why this would be easy to see.