Proving O(1) is the Same as O(2)

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O(1) and O(2) can be considered equivalent in the context of big O notation, as both represent constant time complexity. The discussion highlights that for any constant functions f(n) = c_1 and g(n) = c_2, it holds true that f(n) ≤ g(n) for all n, establishing a relationship between the two. The formal definition of big O notation supports the assertion that any function classified as O(1) is also O(2) and vice versa, reinforcing the concept of "sameness" in terms of growth rates. The Wikipedia article on big O notation is mentioned as a potential resource for further clarification on this topic.
hholzer
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I'm not sure if there is another more appropriate place to ask this question but here it is:
Prove that O(1) is the same as O(2)

O(1) would be the same as O(2) in the sense that for all n, given the constant functions
f(n) = c_1, and g(n) = c_2 we have f(n) <= g(n), yes? Or should "same" be taken more
literally here? If so, then in what way would we prove this?
 
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The Wikipedia article on big O notation may be useful here. I think you could use the formal definition to prove that any function which is O(1) is also O(2), and vice-versa, which seems like a reasonable "sameness condition" to me.
 
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