not injective/not surjective linear maps Let $S$ be the vector space of real sequences, and for $x=(x_1,x_2,\dots)$ define $\alpha(x)=(0,x_1,x_2,\dots)$ and $\beta(x)=(x_2,x_3,\dots)$. The problem was asking for few other things to do, but I got stuck at showing that the first is not injective, while the second is not surjective.
Now, I realize I need to find two distinct arguments (two different sequences), plug it in $\alpha(x)$ and get the same value, which would show that is not injective. But I can not think of any example. Could somebody guide me, how I should structure my search for such example?
As for second, could I take for example a sequence of one member $\{x_1\}$? Then it would be undefined for $\beta$ proving it's not surjective, right?

Hints:
$$\{1,1,...,1,\ldots\}\notin\text{Im}\,(\alpha)\,,\,\,\text{but}\;\;\{0,x_1,x_2,\ldots\}=\{0,y_1,\ldots\}\iff$$
$$ \{x_1,x_2,\ldots\}=\{y_1,y_2,\ldots\}$$
$$\forall\;x:=\{x_1,x_2,...\}\;,\;\;\beta\{0,x_1,x_2,\ldots\}=x\;,\;\;\text{yet}$$
$$\beta\{1,2,3,\ldots\}=\beta\{0,2,3,\ldots\}$$
