closed set, normed spaces

iris_m

1. The problem statement, all variables and given/known data
Let [tex]X=(C([0,1]), || . ||_1 )[/tex], where [tex]||f||_1=\int_{0}^{1}|f(t)|dt[/tex].
Let [tex]M=\{f \in C([0,1]) : \int_{0}^{1}f(t)dt=2, f(1)=0\}[/tex].
Is M closed in X?

3. The attempt at a solution

I've tried the following:
Let [tex]f_n[/tex] be a sequence in M such that [tex]f_n \rightarrow f[/tex].
I'm checking whether [tex]f \in M[/tex], because that would prove that M is closed (if a set contains all the limits of its convergent sequences, it is closed).
There are obviously two conditions to check.
The first one:
[tex]\int_{0}^{1}f(t)dt=\int_{0}^{1}limf_n(t)dt=lim\int_{0}^{1}f_n(t)dt=lim 2=2[/tex].
Now I have to check the second one, that is, is [tex]f(1)=0[/tex], and I don't know how.

Any help is much appreciated.

HallsofIvy

If [itex]f_n(1)= 0[/itex] and [itex]f_n\rightarrow f[/itex], then.... (What kind of convergence are you talking about? Pointwise? Uniform? In the norm?)

And is your justification for saying
[tex]\int_0^1 limf_n(t)dt= lim\int_0^1 f_n(t)dt[/tex]?

iris_m

In the norm.

My justification would be that f_n are continuous functions, so integral and limit commute.