I was wondering if I have two convergent series, say, $\sum_{n=1}^{\infty} s_n = s$ and $\sum_{n=1}^{\infty} t_n$ = t, and for all $n \in \mathbb{N}$ the terms satisfy:
$s_n > t_n$.
Is it then necessary that we have:
$s > t$?
My intuition wants to say yes, but then I know funky stuff goes down with sums, products, intersections, unions and the likes when they deal with infinity.
When dealing with a situation like this I would probably inductively show the partial sums have a positive difference, then assume that the limits of the series must have a positive difference too. However this isn't rigorous at all to me and I was wondering if there was an actual proof (if this is a true implication).
Thanks in advance.
Suppose $s=t$. Since both sums converge, it follows that $\sum_{n=1}^\infty (s_n-t_n)=0$. Since $(s_n-t_n)>0$, $\sum_{n=1}^k(s_n-t_n)$ is monotonically increasing in $k$ which means that $s_n=t_n$ must be true, which is a contradiction. Thus $s=t$ iff $s_n=t_n$.