Why is this Inequality True for all Positive Real Numbers? I was reading Folland's Real Analysis and came across the following inequality in chapter 6 (it is from lemma 6.1)

Let $0<\lambda<1$. Then $t^\lambda\leq \lambda t +(1-\lambda)$, $t$ a
  real positive number.

Is this inequality always true? I'm not sure how to prove it and doesn't seem evident to me. 

If you want to avoid concavity arguments just note that the function $$\phi(t) = t^\lambda - \lambda t - (1-\lambda)$$
has derivative $$\phi'(t) = \lambda t ^{\lambda - 1} - \lambda$$ which is positive if $0 < t < 1$ and negative if $t > 1$. Thus $\phi$ increases to its maximum at $t = 1$ and then decreases so that $$t^\lambda - \lambda t - (1-\lambda) \le \phi(1) = 0$$ for all $t > 0$.
