- #1

- 132

- 0

## Homework Statement

Let f be a continuous function on ℝ. Suppose that [itex]\mathop {\lim }\limits_{x \to - \infty } f(x) = 0[/itex] and [itex]\mathop {\lim }\limits_{x \to \infty } f(x) = 0[/itex]. Prove that there exists a number M > 0 such that [itex]\left| {f(x)} \right| \le M[/itex] for all [itex]x \in ℝ[/itex].

## Homework Equations

[itex]\mathop {\lim }\limits_{x \to - \infty } f(x) = 0[/itex] ⇔ for every ε > 0 there is N such that if x > N then [itex]\left| {f(x)} \right| < ε[/itex]

[itex]\mathop {\lim }\limits_{x \to - \infty } f(x) = 0[/itex] ⇔ for every ε > 0 there is N such that if x < N then [itex]\left| {f(x)} \right| < ε[/itex]

## The Attempt at a Solution

I can see something similar to the precise definition of limits at infinity in the question but I'm not sure if this is the case. Any hint is appreciated, thanks a lot!