The finite number of terms at the beginning of the sequence are bounded between the maximum of

and the minimum of

for

. Therefore these finite number of terms are already bounded and it is only necessary to prove that the infinite number of terms that follow (

) are bounded.

This is where I think the question essentially answers itself. From the above argument, the finite number of terms at the start are bounded and the question states that the sequence is "eventually bounded" so all the infinite number of terms for n>N are also bounded. Hence the entire sequence is bounded.