Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P\in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $PQ$, where $Q\in Y$, $Q\ne P$. Show that $\operatorname{deg} X<d$.

This problem appears in Hartshorne's book under the guise of ex. I.7.7b. Though not research level, asking around at my home institution hasn't turned anything up, and math.stackexchange has proven a graveyard. I would like to prove this statement in the following way but am missing one ingredient; I was hoping that perhaps someone could put me out of my misery. Of course, if this question is inappropriate for this site, feel free to close, etc. (This is not homework.)

Note that $X$ is a variety of degree $r+1$. Suppose first $r=1$. Then take a hyperplane $H$ containing $P$ and $d-1$ other points on $Y$, counted with multiplicity. Then $H\cap X$ contains at most $d-1$ lines, counted without multiplicity. We would be done with this case if we could conclude that a given irreducible component (line) $L$ in $H\cap X$ has an intersection multiplicity $i$ which is less than or equal to the number of points on $L\cap Y$ other than $P$, counted with multiplicity.

Now suppose $r>1$. Take a hyperplane $H$ containing $P$ and intersect with $X$ and $Y$. Then both decrease in dimension by one, and $X\cap H$ is the closure of the union of all lines from $P$ to points in $Y\cap H$. Hence one can use the inductive hypothesis to conclude that $X\cap H$ has lesser degree than $Y\cap H$. To be done, one needs some guarantee that the intersection multiplicity of any irreducible component $Z$ of $H\cap X$ is less than or equal to the sum of the intersection multiplicities of the irreducible components of $H\cap Y$ contained in $Z$.

I am unable to complete the proof since I am missing the statement 'an irreducible component $Z$ of $H\cap Z$ has intersection multiplicity less than or equal to the sum of the intersection multiplicities of the irreducible components of $H\cap Y$ contained in $Z$.'

Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P\in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $PQ$, where $Q\in Y$, $Q\ne P$. Show that $\operatorname{deg} X<d$.

The comments below resolve this problem.