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square both sides and simplify
\[ ac+bd \le \sqrt{a ^{2}+b ^{2}}\sqrt{c ^{2}+d ^{2}}\]
square both sides and simplify
\[2abcd \le b ^{2}c ^{2} + a ^{2}d ^{2}\]
make substitution
u =bc
v = ad
\[2uv \le u ^{2}+v ^{2}\]
look at all 3 cases,
u=v
2u^2 <= 2u^2
True
u>v: u=v+1
\[2v(v+1)\le v ^{2}+(v+1)^{2}\]
\[\rightarrow 2v ^{2}+2v \le 2v ^{2}+2v+1\]
0 <= 1
True
u

I'll be back later to collect my medals.

lol

here

Unfortunately, you gave one to the first post too, so that is unacceptable.

I thought you were harsh.