# The space $(C[a,b],\rho_{p})$ is Incomplete proof

I have difficulty understanding this proof of Shilov's book, Elementary Functional Analysis

Here is my trial to understand the proof. Given an $$\epsilon > 0$$ and a sequence $$\{ y_n\}$$ where $$y_n(x) \rightarrow 0$$ uniformally on $$[a,c-\epsilon]$$ and $$y_n(x) \rightarrow 1$$ uniformly on $$[c+\epsilon,b]$$ then, consider the distance between $$y_m(x)$$ and $$y_n(x)$$ and for sufficiently large $$m,n$$ we have $$\rho^{p}(y_m(x),y_n(x)) = \int_{a}^{c-\epsilon} 0 dx +\int_{c-\epsilon}^{c+\epsilon} | y_m(x)-y_n(x)|^{p} dx + \int_{c+\epsilon}^{b}0 dx = \int_{c-\epsilon}^{c+\epsilon} | y_m(x)-y_n(x)|^{p} \leq \int_{c-\epsilon}^{c+\epsilon} 1 dx = 2\epsilon$$ then indeed $$\{y_n(x)\}$$ is a cauchy sequence on $$[a,b]$$.

Question. Why $$\rho^{p}(y_m(x),y_n(x)) \leq 4\epsilon$$?

• $y_m,y_n$ need not equal on $[a,c-\epsilon]$ (and $[c+\epsilon,b]$), so you need two $\epsilon$s to help you there. – user10354138 Nov 9 '18 at 9:29

I got that in the proof he said that $$y_n(x)$$ is a sequence of functions taking values between $$0$$ and$$1$$,that is for each $$n\in N$$ and $$x\in[a,b]$$ we have $$0\leq y_n(x)\leq 1$$. therefore $$|y_n(x)-y_m(x)|\leq 1$$. Now since the sequence $${y_n(x)}$$ is a converges sequence in $$[a,c-e]$$ and$$[c+e,b]$$ ,then there is a natural number $$N$$ such that for each $$n,m\geq N$$ and $$x \in [a,c-e]$$ or $$x\in [c+e,b]$$ we have $$|y_n(x)-y_m(x)| \lt ({\frac{e}{b-a}})^\frac 1p=T$$. Hence $$\int_{a}^{c-e}|y_n(x)-y_m(x)|^p \lt \int_{a}^{c-e}{T}^p=\int_{a}^{c-e}\frac{e}{b-a}=\frac{e}{b-a}((c-e)-a) \leq \frac{e}{b-a}(b-a)=e$$ Similarlly we have$$\int_{c+e}^{b}|y_n(x)-y_m(x)| \lt e$$. Hence $$\int_a^b |y_n(x)-y_m(x)| \lt e+2e+e=4e$$