Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices X,Y it holds that ||e^{X+Y}-e^{X}|| \leq ||Y||e^||X|| e^||Y|| where ‖ · ‖ denotes an arbitrary matrix norm. I am trying to prove it since I cannot find it as a reference somewhere. Using the series I cant say that the result is the desired one.

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ denotes an arbitrary matrix norm. I am trying to prove it since I cannot find it as a reference somewhere. Using the series I can't say that the result is the desired one.