Metamath Proof Explorer


Theorem sqrtlim

Description: The inverse square root function converges to zero. (Contributed by Mario Carneiro, 18-May-2016)

Ref Expression
Assertion sqrtlim
|- ( n e. RR+ |-> ( 1 / ( sqrt ` n ) ) ) ~~>r 0

Proof

Step Hyp Ref Expression
1 rpcn
 |-  ( n e. RR+ -> n e. CC )
2 cxpsqrt
 |-  ( n e. CC -> ( n ^c ( 1 / 2 ) ) = ( sqrt ` n ) )
3 1 2 syl
 |-  ( n e. RR+ -> ( n ^c ( 1 / 2 ) ) = ( sqrt ` n ) )
4 3 oveq2d
 |-  ( n e. RR+ -> ( 1 / ( n ^c ( 1 / 2 ) ) ) = ( 1 / ( sqrt ` n ) ) )
5 4 mpteq2ia
 |-  ( n e. RR+ |-> ( 1 / ( n ^c ( 1 / 2 ) ) ) ) = ( n e. RR+ |-> ( 1 / ( sqrt ` n ) ) )
6 1rp
 |-  1 e. RR+
7 rphalfcl
 |-  ( 1 e. RR+ -> ( 1 / 2 ) e. RR+ )
8 cxplim
 |-  ( ( 1 / 2 ) e. RR+ -> ( n e. RR+ |-> ( 1 / ( n ^c ( 1 / 2 ) ) ) ) ~~>r 0 )
9 6 7 8 mp2b
 |-  ( n e. RR+ |-> ( 1 / ( n ^c ( 1 / 2 ) ) ) ) ~~>r 0
10 5 9 eqbrtrri
 |-  ( n e. RR+ |-> ( 1 / ( sqrt ` n ) ) ) ~~>r 0