Is this correct?

If $$\lim_{x \to a} f_1(x) = L_1, \lim_{x \to a} f_2(x) = L_2, ..., \lim_{x \to a} f_n(x) = L_n$$ Then $$\lim_{x \to a} \sum_{k=1}^n f_k(x) = \sum_{k=1}^n L_k$$


We must prove that: $$\forall \epsilon \gt 0, \exists \delta \gt 0 \; \bigl| \; 0 \lt | x - a | \lt \delta \implies |\sum_{k=1}^n f_k(x) - \sum _{k=1}^n L_k| \lt \epsilon$$

Now for any $\epsilon \gt 0$, consider $\frac \epsilon n$. Then we have: $$\delta_1 \gt 0 \; \bigl| \; 0 \lt |x - a| \lt \delta_1 \implies |f_1(x) - L_1|\lt \frac \epsilon n \\ \delta_2 \gt 0 \; \bigl| \; 0 \lt |x - a| \lt \delta_2 \implies |f_2(x) - L_2|\lt \frac \epsilon n \\ \cdot \\ \cdot \\ \cdot \\ \delta_n \gt 0 \; \bigl| \; 0 \lt |x - a| \lt \delta_n \implies |f_n(x) - L_n|\lt \frac \epsilon n$$

Considering $\delta = min\{\delta_k\}_{k = 1}^n$ we have that:

$$\delta=min\{\delta_ k\}_{k = 1}^n \; \bigl| \; 0 \lt | x - a | \lt \delta \implies \sum_{k = 1}^n|f_k(x) - L_k| \lt \sum_{k = 1}^n \frac \epsilon n = \epsilon$$

But because of the triangle inequality, it holds that: $$\sum_{k = 1}^n|f_k (x)- L_k| \ge |\sum_{k = 1}^n f_k(x) - L_k| = |\sum_{k = 1}^n f_k(x) - \sum_{k = 1}^n L_k|$$

Therefore: $$\delta=min\{\delta_ k\}_{k = 1}^n \; \bigl| \; 0 \lt | x - a | \lt \delta \implies |\sum_{k = 1}^n f_k(x) - \sum_{k = 1}^n L_k| \lt \epsilon$$

  • $\begingroup$ It's a bit superfluous to prove this for arbitrary $n$ rather than simply for $n=2$, but yes, this is the idea; it just comes down to the triangle inequality. $\endgroup$ Aug 13 '16 at 1:57
  • $\begingroup$ So is the triangle inequality in that case right? My greatest doubt was in there. A professor of mine said the symbol was supposed to be inverted, that is: $\le$ instead of $\ge$. $\endgroup$ Aug 14 '16 at 17:11

Yes, this is right, though it could have been written up more carefully. For example, the notation $\delta_1\,|\,...$ doesn't mean anything. What you meant is $\color{red}{\exists}\delta_1\,|\,...$.

In the comments you've asked about your use of the triangle inequality. The key point is that


using the fact that $|a+b|\leq|a|+|b|$. The hypotheses give bounds on the righthand side, hence a bound on the lefthand side.

What you wrote is correct, but perhaps your professor read too quickly and expected you to write the inequality in the other direction. That is, you wrote (the $n$-fold generalization of)


but then you used this fact in the opposite direction. So it would have been more natural to write it the other way.


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