From this question, I'm honestly not too sure on how to prove it.
I've thought about an approach, which is to take 3 numbers from t. Then prove that the number in the middle of the product of those 3 numbers and n contains a 7.
e.g.
Let the t be 3.14159, we take the number "$415$".
Realize that $415\cdot5=2075$, the product contains the number "$7$".
We can also see that if we add any other number in front or behind the number "$415$", the "$07$" in the middle would not be disturbed or changed.
Thus proving that there is a positive integer $n$ such that the decimal expansion of $nt$ contains a $7$ in this particular case.
So the question can be proved as long as prove that all 3 digit numbers have a positive integer such that the product contains a $7$.
However, now I'm not sure on how to approach this anymore.
I'm also not sure if this approach is possible.
Can someone help me, thank you.