Final answer:
To prove the statement, we can use the fact that if a function is continuous on a closed interval, then it is also bounded on that interval. By considering the interval [-R, R] for some positive R, we can show that f is bounded on the entire ℝ.
Step-by-step explanation:
To prove the statement, we will use the fact that if a function is continuous on a closed interval, it is also bounded on that interval. Since we are given that f is continuous on ℝ, we can consider the interval [-R, R] for some positive real number R. By the definition of limits, for any ε > 0, there exists a positive real number M such that for all x > R, |f(x) - M| < ε. Similarly, there exists a negative real number N such that for all x < -R, |f(x) - N| < ε. Therefore, f is bounded on the interval [-R, R], and since R can be arbitrarily large, f is bounded on the entire ℝ.