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Let $G=(V,T,P,S)$ be a context-free grammar without a production rule for $epsilon$. Define a new context-free grammar $G'$ which produces every word in $L(G)$ such that the word is without its last letter:
$$
L(G')=bigg{win T^*bigg|exists tin T: wtin L(G) bigg}
$$

The solution to the problem is as follows:
let $G'=(Vcup hat V, T,P', hat S)$, where $hat V$ is the set of variables in $V$ but with the hat sign. For, all $Ain V$:
$$
ifquad (Ato alpha)in P implies (Ato alpha)in P'\
ifquad(Ato alpha t)implies (hat Ato alpha)in P'quad (*)\
ifquad(Ato alpha X)in P, Xin Vimplies (hat Ato alpha hat X)in P'quad (**)
$$

I think I understand the solution: if some production rule leads to a letter which is not the last, we add the rule as is to $P'$. If the rule leads to a string whose last letter is $t$ then we add the string without $t$ to $P'$. Lastly, if a rule leads to a variable and a letter we add them to $P'$.

I don't understand why we needed to define $hat V$. I guess I could understand its use in the $2$nd case (*) because if $Ato alpha t$ then $Ato alpha$ didn't really exist in $V$. But then I don't understand why we need to "hat-ify" the last case (**). $Atoalpha X$ was in $V$ so it continues to exist in $V$ why do we need to add it as $hat Ato alpha hat X$?