Proof that $\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$

Question

I am looking for an analytical proof that : $$\sum_{y=0}^{\infty} \frac{x^y y}{y!} = xe^{x}$$

Both CAS Wolfram alpha and sympy agree on the result :

from sympy import I, oo, Sum, exp, pi, factorial
from sympy.abc import y,x
print(Sum(y*x**y / factorial(y),(y,0,oo)).doit())
# prints x*exp(x)

but do not provide proof. I tried few things like substitution, exp/ln rewritting but ended up on undefined $1/factorial(-1)$

Answer 1

$$\sum_{k\ge0}\frac{x^k k}{k!}=\sum_{k\ge1}\frac{x^k}{(k-1)!}=x\sum_{k\ge1}\frac{x^{k-1}}{(k-1)!}=x\sum_{n\ge0}\frac{x^n}{n!}=xe^x.$$