# fa.functional analysis – false proof for a.s convergence of Sub-martingale

Let $$f: mathbb {R} to mathbb{R}$$ be a right-continuous and such that for all $$p in mathbb {N}^*,$$ the sequence $$(f (frac {k} {2^p}))_{k inmathbb{N}}$$ converges in $$mathbb {R}.$$ Can we deduce that $$f$$ admits a limit in $$+ infty.$$ I think no, it’s sufficient to take $$displaystyle f(x)= sum _{n=0}^infty gBig( 4^n Big( x-Big(frac32Big)^n Big) Big)$$ where g is defined by
$$g(x)= 1-x$$ if $$x in (0,1), g(x)= 1+x$$ if $$x in (-1,0($$ and $$g(x)=0$$ if $$|x|>1$$. notice that $$f$$ does not admit a limit in $$+ infty.$$ and for all $$p in mathbb {N}^*,$$ the sequence $$(f (frac {k} {2^p}))_{k inmathbb{N}}$$ converges to 0

Finally, it seems to me that this proof is false