How can I prove $AX=BX$ for every $n\times1$ column matrix $X \implies A=B$

Let $e_i$ denote the $i^{th}$ standard basis vector. Then for all $1\leq i \leq n$,

$$Ae_i = Be_i.$$

But $Ae_i$ and $Be_i$ are just the $i^{th}$ columns of $A$ and $B$ respectively.

Consider the column vectors $e_i$ which are all $0$s except for the $i$th component. Then $Ae_i$ is the $i$th column of $A$, which is the same as the $i$th column of $B$, $Be_i$. Thus, all the columns in $A$ equal the corresponding columns in $B$.