Finite sequences of unbounded value

Question

If I have a finite sequence of expressions $a_1+a_2+a_3+....a_k=\infty$, does that imply that at least one such $a_j=\infty$?

I know that if it didn't it would make the sum not equal to infinity, but it still does not make intuitive sense to me. Couldn't part of the magnitude be contained in each expression equally and still be such that one term didn't diverge?

Answer 1

Asking whether $a_1+...+a_k=\infty$ implies at least one of the $a_i$ is $\infty$ only makes sense if we have defined addition of $\infty$. If we work with real numbers only then the sum of a finite number of real numbers is certainly a real number.

Answer 2

Put it this way: the sum of $k$ finite expressions is finite. Prove by induction on $k$.

Answer 3

Yes; suppose that is not the case. Then we know $a_i\leq\max\{a_1,a_2,\ldots,a_k\}=M<\infty$ by hypothesis. Then $\sum a_i\leq k\cdot M<\infty$, a contradiction. Thus, at least one must be infinity.