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## Homework Statement

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?

## 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.