injective functions always have inverse True or False

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" Injective functions always have inverse " statement is false.

This statement is only true for finite dimension space, since in finite dimensional space every injective function is onto and hence invertible.

In infinite dimensional space, let the function is

$f(x_1,x_2,x_3, x_4,....) = (0,x_1, x_2,x_3,x_4,....)$ .

This function is injective, since $f(X)=f(Y) \implies X=Y$ .

But function is onto, since co-domain element $(1,x_1,x_2,x_3,x_4, ...)$ has no pre-image.

Hence, this function is injective but not invertible.

Thus, " Injective functions always have inverse " statement is false.