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Continuity & sequences

Question:

Suppose f:R→R f: \mathbb{R} \rightarrow \mathbb{R} is a function satisfying:
lim⁡x→+∞f(x)=lim⁡x→−∞f(x)=−∞\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty

part a) Show that if (xn)(x_n) is any sequence of real numbers such that the sequence (f(xn))(f(x_n))either converges to a real number or diverges to +∞+\infty, then (xn)(x_n) is bounded.

my attempt of a solution of part a)

All I know so far is that for some value A, B we have x<A⇒f(x)<M x<A \Rightarrow f(x) < M and x>B⇒f(x)<M x>B \Rightarrow f(x) <M fromlim⁡x→+∞f(x)=lim⁡x→−∞f(x)=−∞\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty

but I really don't know what to do with this - looking around I think that f(x_n) is bounded below, but still - I don't know where to proceed
Original post by zoxe
Question:

Suppose f:R→R f: \mathbb{R} \rightarrow \mathbb{R} is a function satisfying:
lim⁡x→+∞f(x)=lim⁡x→−∞f(x)=−∞\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty

part a) Show that if (xn)(x_n) is any sequence of real numbers such that the sequence (f(xn))(f(x_n))either converges to a real number or diverges to +∞+\infty, then (xn)(x_n) is bounded.

my attempt of a solution of part a)

All I know so far is that for some value A, B we have x<A⇒f(x)<M x<A \Rightarrow f(x) < M and x>B⇒f(x)<M x>B \Rightarrow f(x) <M fromlim⁡x→+∞f(x)=lim⁡x→−∞f(x)=−∞\displaystyle \lim_{x\rightarrow +\infty} f(x) = \lim_{x\rightarrow -\infty} f(x) = -\infty

but I really don't know what to do with this - looking around I think that f(x_n) is bounded below, but still - I don't know where to proceed

Try a contradiction i.e. suppose (x_n) is unbounded and then consider f([a divergent subsequence]).
(edited 10 years ago)

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