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In a book by Charalambos D. Aliprantis and Owen Burkinshaw, Positive Operators, in page 13, Example 1.16, there is a statement which I cannot see why it is always true. The statement is as follows:

Take $0<f∈C[−1, 1]$, and let $0<c<2π$. Also, for each $n∈{\mathbb{N}}$, let $\displaystyle t_n = \frac{1}{c+2nπ}$ and note that $t_n → 0$. Next pick some $g_n ∈ C[−1, 1]$ with $0≤g_n≤f$ such that $g_n(\text{sin} \ c)=f(\text{sin} \ c)$ and $g_n(\text{sin}(c+t_n))= 0$.

Now, my question is: Which theorem (or result) guarantees that, for each $n∈{\mathbb{N}}$, there is a function $g_n$ with these properties? (Is it an application of Urysohn Lemma? If so, how? (Here $C[−1, 1]$ denotes the vector space of continuous functions on the interval $[-1,1]$.)

Can anybody help me to understand/know this passage in the above aforementioned example?

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Let $h_n$ be any non-negative continuous function such that $h_n(\sin(c+t_n))=0$ and $h_n(\sin c)=f(\sin c)$, for instance $$ h_n(x)=\frac{|x-\sin(c+t_n)|}{|\sin c-\sin(c+t_n)|}\,\,f(\sin c). $$ Then take $g_n(x)=\min(f(x),h_n(x))$.

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