Exercise $1.$ One way that a Turing machine $M$ can fail to terminate on a string $w$ is that, for all ninN, the program will at some stage have more than $n$ symbols on the tape. That is$,$ the output grows without bound. Call this infinite extension.

Let $A_{IE}$ be the language of pairs $($:$M,w$:$)$ where $M$ is a Turing machine and $w$ is a string such that the program accepts if and only if $M$ runs an infinite extension on $w.$ Prove that $A_{IE}$ is undecidable.

$($Hint: This can be done by reducing this to $A_{HT.})$