Let $f:X\rightarrow \mathbb{R}$ be a convex function. Show that for any $x,y\in X,0<\mu<\lambda$, we have $$\frac{f(x+\mu y)-f(x)}{\mu}\le\frac{f(x+\lambda y)-f(x)}{\lambda} $$
My proof first shows that $x = \frac{\lambda}{\lambda-\mu}(x+\mu y)-\frac{\mu}{\lambda-\mu}(x+\lambda y)$. Then, \begin{align*}f(x) &= f\left(\frac{\lambda}{\lambda-\mu}(x+\mu y)-\frac{\mu}{\lambda-\mu}(x+\lambda y)\right)\\ & \le \frac{\lambda}{\lambda-\mu}f\left((x+\mu y)\right)-\frac{\mu}{\lambda-\mu} f\left((x+\lambda y)\right)\\ (\lambda-\mu)f(x) &\le \lambda f\left((x+\mu y)\right)-\mu f\left((x+\lambda y)\right)\\ \mu(f\left((x+\lambda y)\right)-f(x)) &\le \lambda(f\left((x+\mu y)\right)-f(x))\\ \frac{f(x+\lambda y)-f(x)}{\lambda}&\le\frac{f(x+\mu y)-f(x)}{\mu}\end{align*}
Which I get different from what question asks. At which point I made a mistake?? Thanks,