Let $X$ be a non negative random variable. Show that $$\sum_{n=1}^\infty P\{X\ge n\}\le EX\le 1+\sum_{n=1}^\infty P\{X\ge n\}.$$ Now, the thing is I need this to prove the next part of the question, which is $EX=\int_0^\infty P\{X\ge x\}\lambda(dx)$, so I am not supposed to use this definition for proving the above inequality. I have managed to prove the following by breaking up the integrals but, I'm stuck after it. $$\sum_{n=1}^\infty\int_n^{n+1}xf(x)\lambda(dx)\le EX\le1+\sum_{n=1}^\infty\int_n^{n+1}xf(x)\lambda(dx)$$ Somehow proving $\sum_{n=1}^\infty\int_n^{n+1}xf(x)\lambda(dx)=P\{X\ge n\}$ should solve my issue but I don't think it to be true. Perhaps there is a different approach? Help would be great, please.
NOTE. There are similar questions on the site but with answers I'm unable to completely understand or with some modifications like X only takes integral values etc.