algorithm - Comparing the Bounds of Functions -

i understand general idea of bounds of functions; example, if function

ƒ1(n) ∈ ω(n^2)

then know ƒ1(n) element within constraints of lower bound n^2, meaning ƒ1(n) can function grows slower or equal of n^2.

now start confused when talk other functions in regards bounds of ƒ1(n). example, have statement claims this:

if

ƒ1(n) ∈ ω(n^2)

and

ƒ2(n) ∈ θ(n)

then

ƒ2(n) ∈ o(ƒ1(n))

i have hard time telling whether or not it's true or false. have 2 different approaches contradict each other:

• true - since ƒ2 tightly bound under n, can considered element within constraints of o(ƒ1(n)) because ƒ2 not grow slower ƒ1.
• false - since ƒ1 has lower bound of n^2, function ƒ2, tightly bound under n, cannot considered element of upper bounds of ƒ1 since know ƒ1 not have upper bound grows slower n^2.

both of these approaches i've thought of seem valid me; think i'm getting confused on whether or not care lower bounds of ƒ1 when we're trying decide if function element of upper bounds.

any on clarifying appreciated.

let's @ definitions of big-o etc. formulas valid sufficiently large n. then, there exist constants k, such that:

first formula:

ƒ1(n) ∈ Ω(n^2)
⇔ k1 * n^2 <= f1(n)

second formula:

ƒ2(n) ∈ Θ(n)
⇔ k2 * n <= f2(n) <= k3 * n

we know k3 * n < k1 * n^2 (again, sufficiently large n). hence:

k2 * n <= f2(n) <= k3 * n < k1 * n^2 <= f1(n)

this can reduced to

f2(n) < f1(n)

with knowledge, see inferred formula true:

ƒ2(n) ∈ o(ƒ1(n))
⇔ f2(n) <= k4 * f1(n)