evinda
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Hey again!
(Blush)
I am looking at the following exercise:
Let $f:[0,+\infty) \to \mathbb{R} $ integrable on each closed subinterval of $[0,+\infty)$.We suppose that $\lim_{x \to +\infty} f(x)=l$.Show that $\lim_{x \to +\infty} \frac{1}{x} \int_{0}^{x} f = l$.
The solution is the following:
Let $\epsilon>0$.As $\lim_{x \to +\infty} f(x)=l$, $\exists c_1>0$ such that $|f(x)-l|<\epsilon, \forall x \geq c_1$.
Also,$\exists c_2>0$ such that $\frac{1}{x} \int_{0}^{c_1} |f-l|< \epsilon, \forall x \geq c_2$
Therefore, $ \forall x \geq \max\{c_1,c_2\}$ we have $|\frac{1}{x} \int_0^x f - l|=\frac{1}{x} |\int_0^x (f-l)|\leq \frac{1}{x} \int_0^{c_1} |f-l|+\frac{1}{x} \int_{c_1}^x |f-l| \leq \epsilon + \frac{x-c_1}{x} \epsilon \leq \epsilon$
Could you explain me the red part?? How do we conclude to that??

I am looking at the following exercise:
Let $f:[0,+\infty) \to \mathbb{R} $ integrable on each closed subinterval of $[0,+\infty)$.We suppose that $\lim_{x \to +\infty} f(x)=l$.Show that $\lim_{x \to +\infty} \frac{1}{x} \int_{0}^{x} f = l$.
The solution is the following:
Let $\epsilon>0$.As $\lim_{x \to +\infty} f(x)=l$, $\exists c_1>0$ such that $|f(x)-l|<\epsilon, \forall x \geq c_1$.
Also,$\exists c_2>0$ such that $\frac{1}{x} \int_{0}^{c_1} |f-l|< \epsilon, \forall x \geq c_2$
Therefore, $ \forall x \geq \max\{c_1,c_2\}$ we have $|\frac{1}{x} \int_0^x f - l|=\frac{1}{x} |\int_0^x (f-l)|\leq \frac{1}{x} \int_0^{c_1} |f-l|+\frac{1}{x} \int_{c_1}^x |f-l| \leq \epsilon + \frac{x-c_1}{x} \epsilon \leq \epsilon$
Could you explain me the red part?? How do we conclude to that??
