Can a sequence contain an infinite number of numbers 1 or above and still converge to a non 1 value? Can we have a sequence $a_n$ that has an infinite amount of 1's, yet still converge to a number $\neq$ 1? 
I know that the sequence of $(-1)^n$ from 1 to $\infty$ does not converge, but can this be remedied if we exclude the negative numbers? 

Nope. If an infinite sequence has a limit $L$, then any infinite subsequence has the same limit $L$.
