What is wrong with this proof that \( 2 = 4 \)?
Step A:
Let \({ x }^{ {\displaystyle x }^{ {\displaystyle x }^{\displaystyle .^{.^.} } } }=y\).
Substituting, we get \({ x }^{\displaystyle y }=y\)
Step B:
\(x=\sqrt [ y ]{ y } \). Letting \( y = 4 \), then \( x = \sqrt[4]{4}\) and hence
\({ \sqrt [ 4 ]{ 4 } }^{ {\displaystyle \sqrt [ 4 ]{ 4 } }^{ {\displaystyle \sqrt [ 4 ]{ 4 } }^{\displaystyle .^{.^.} } } }=4\)
Step C:
\(x=\sqrt [ y ]{ y } \). Letting \( y = 2 \), then \( x = \sqrt[2]{2}\) and hence
\({ \sqrt [ 2 ]{ 2 } }^{ {\displaystyle \sqrt [ 2 ]{ 2 } }^{ {\displaystyle \sqrt [ 2 ]{ 2 } }^{\displaystyle .^{.^.} } } }=2\)
Step D:
Observe that \(\sqrt { 2 } = 2^ \frac{1}{2} = \sqrt [ 4 ]{ 4 } \). Hence, we get
\[ 4 = { \sqrt [ 4 ]{ 4 } }^{ {\displaystyle \sqrt [ 4 ]{ 4 } }^{ {\displaystyle \sqrt [ 4 ]{ 4 } }^{\displaystyle .^{.^.} } } }= \sqrt [ 2 ]{ 2 } ^{ {\displaystyle \sqrt [ 2 ]{ 2 } }^{ {\displaystyle \sqrt [ 2 ]{ 2 } }^{\displaystyle .^{.^.} } } } = 2 \]
Which step is wrong?
Note: \({ x }^{ {\displaystyle x }^{ {\displaystyle x }^{\displaystyle .^{.^.} } } }=y\) is an infinite tetration, or infinite tower of exponentiation