Proving exponential inequalities

Question

I'm currently revising for an upcoming exam and am stuck on the following question. I have completed a similar question that involved cos and the mean value theorem I used the triangle inequality too, so I assume this question will do the same. However I am unsure how to go about this. Any help would be appreciated.

Answer 1

Hint: According to the MVT $$e^x-e^y=e^\xi (x-y) $$ for some $\xi \in (0,1).$

Answer 2

One may observe that, for $x,y \in \mathbb{R}$, $$ e^x-e^y=\int_x^ye^tdt $$ giving, for $x,y \in [0,1]$, $$ \left|e^x-e^y\right|=\left|\int_x^ye^tdt\right|\leq e\left|\int_x^y1dt\right|=e|x-y|. $$