How $\lim_{x \to \infty} f(x)/x=1$ can be derived from $1/\lambda<f(x)/x<\lambda$?

Question

Could any one please elaborate how $\lim_{x \to \infty} f(x)/x=1$ can be derived from $1/\lambda<f(x)/x<\lambda$? I have taken it granted that $1/\lambda<f(x)/x<\lambda$.

The question is taken from below proof-

PS: I made a mistake in previous post so, I am re-posting.

Answer 1

Pay close attention to the following fragment of the proof: "These inequalities hold for all $\lambda \gt 1$." In particular, consider $\lambda$ arbitrarily close to the value one.