Let $f_n : [0,\infty] \rightarrow [0,\infty]$ be sequence of functions such that $f_n(t)$ converges to linear function $ht$ as $n\rightarrow \infty$ for some $h>0$ for all $t\in[0,\infty]$. Define, for $p>0$ fixed, $\tau_n = \inf\{t\geq0:f_n(t)>p\}.$ Show that $\tau_n$ converges to $\frac{p}{h}$ as $n\rightarrow \infty$.
If sequence of functions converges, then first crossing time of each function also converges
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sequences-and-series
functions
convergence
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0Any thoughts on the problem? – 2017-02-22
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0Unfortunately, no thoughts or ideas to approach. – 2017-02-22
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0Assume further that $f_n$ is non-decreasing function. – 2017-02-24