For an integer n, let fn(x) = xe−nx, where x ∈ [0, 1]. Let S := {fn : n ≥ 1}. Consider the metric space (C([0, 1]), d), where
\(\rm \displaystyle d(f, g)=\sup _{x \in[0,1]}\{|f(x)-g(x)|\}\), f, g ∈ C([0, 1]).
Which of the following statement(s) is/are true?
1
S is an equi-continuous family of continuous functions
2
S is closed in (C([0, 1]), d)
3
S is bounded in (C([0, 1]), d)
4
S is compact in (C([0, 1]), d)